- Preferences

## Unions and Intersections of Sets - PowerPoint PPT Presentation

- Unions and Intersections of Sets

## The union of two events A and B is the event that occurs if either A or B or ... Demo -Union of Two Events. Intersection of Two Events ... – PowerPoint PPT presentation

- A compound event is the composition of two or more other events. Compound events may be formed by the union or the intersection of two events.
- The union of two events A and B is the event that occurs if either A or B or both occur on the performance of the experiment. A U B consists of all sample points that are in either A or B or both.
- Demo -Union of Two Events
- The intersection of two events A and B is the event that occurs if both A and B occur on a single performance of the experiment. We write AnB for the intersection of a and B. AnB consists of all sample points that belong to both A and B.
- Demo _Intersection of Two Events
- The complement of event A is the event that A does not occur that is, the event consisting of all sample points that are not in A. We denote the complement of A as A' or as A
- Demo of A complement

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## 4.3: Unions and Intersections

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- Harris Kwong
- State University of New York at Fredonia via OpenSUNY

We can form a new set from existing sets by carrying out a set operation.

## Definition: \(A \cap B\)

Given two sets \(A\) and \(B\), define their intersection to be the set

\[A \cap B = \{ x\in{\cal U} \mid x \in A \wedge x \in B \}\]

Loosely speaking, \(A \cap B\) contains elements common to both \(A\) and \(B\).

## Definition: \(A \cup B\)

The union of \(A\) and \(B\) is defined as

\[A \cup B = \{ x\in{\cal U} \mid x \in A \vee x \in B \}\]

Thus \(A \cup B\) is, as the name suggests, the set combining all the elements from \(A\) and \(B\).

UNION

## Definition: \(A-B\)

The set difference \(A-B\), sometimes written as \(A \setminus B\), is defined as

\[A- B = \{ x\in{\cal U} \mid x \in A \wedge x \not\in B \}\]

In words, \(A-B\) contains elements that can only be found in \(A\) but not in \(B\). Operationally speaking, \(A-B\) is the set obtained from \(A\) by removing the elements that also belong to \(B\).

## Definition: \(\overline{A}\)

The complement of \(A\), denoted by \(\overline{A}\), \(A'\) or \(A^c\), is defined as

\[\overline{A} = \{ x\in{\cal U} \mid x \notin A \}\]

## Definition: \(A \bigtriangleup B\)

The symmetric difference \(A \bigtriangleup B\), is defined as

\[A \bigtriangleup B = (A - B) \cup (B - A)\]

## Definition: Disjoint

Two sets are disjoint if their intersection is empty.

For example, consider \(S=\{1,3,5\}\) and \(T=\{2,8,10,14\}\).

\(S \cap T = \emptyset\) so \(S\) and \(T\) are disjoint.

We would like to remind the readers that it is not uncommon among authors to adopt different notations for the same mathematical concept. Likewise, the same notation could mean something different in another textbook or even another branch of mathematics. It is important to develop the habit of examining the context and making sure that you understand the meaning of the notations when you start reading a mathematical exposition.

Example \(\PageIndex{1}\label{eg:unionint-01}\)

Let \({\cal U}=\{1,2,3,4,5\}\), \(A=\{1,2,3\}\), and \(B=\{3,4\}\). Find \(A\cap B\), \(A\cup B\), \(A-B\), \(B-A\), \(A\bigtriangleup B\), \(\overline{A}\), and \(\overline{B}\).

We have \[\begin{aligned} A\cap B &=& \{3\}, \\ A\cup B &=& \{1,2,3,4\}, \\ A - B &=& \{1,2\}, \\ B \bigtriangleup A &=& \{1,2,4\}. \end{aligned}\] We also find \(\overline{A} = \{4,5\}\), and \(\overline{B} = \{1,2,5\}\).

hands-on exercise \(\PageIndex{1}\label{he:unionint-01}\)

Let \({\cal U} = \{\mbox{John}, \mbox{Mary}, \mbox{Dave}, \mbox{Lucy}, \mbox{Peter}, \mbox{Larry}\}\), \[A = \{\mbox{John}, \mbox{Mary}, \mbox{Dave}\}, \qquad\mbox{and}\qquad B = \{\mbox{John}, \mbox{Larry}, \mbox{Lucy}\}.\] Find \(A\cap B\), \(A\cup B\), \(A-B\), \(B-A\), \(\overline{A}\), and \(\overline{B}\).

hands-on exercise \(\PageIndex{2}\label{he:unionint-02}\)

If \(A\subseteq B\), what would be \(A-B\)?

Example \(\PageIndex{2}\label{eg:unionint-02}\)

The set of integers can be written as the \[\mathbb{Z} = \{-1,-2,-3,\ldots\} \cup \{0\} \cup \{1,2,3,\ldots\}.\] Can we replace \(\{0\}\) with 0? Explain.

hands-on exercise \(\PageIndex{3}\label{he:unionint-03}\)

Explain why the following expressions are syntactically incorrect.

- \(\mathbb{Z} = \{-1,-2,-3,\ldots\} \cup \;0\; \cup \{1,2,3,\ldots\}\).
- \(\mathbb{Z} = \ldots,-3,-2,-1 \;\cup\; 0 \;\cup\; 1,2,3,\ldots\,\)
- \(\mathbb{Z} = \ldots,-3,-2,-1 \;+\; 0 \;+\; 1,2,3,\ldots\,\)
- \(\mathbb{Z} = \mathbb{Z} ^- \;\cup\; 0 \;\cup\; \mathbb{Z} ^+\)

How would you fix the errors in these expressions?

Example \(\PageIndex{3}\label{eg:unionint-03}\)

For any set \(A\), what are \(A\cap\emptyset\), \(A\cup\emptyset\), \(A-\emptyset\), \(\emptyset-A\) and \(\overline{\overline{A}}\)?

It is clear that \[A\cap\emptyset = \emptyset, \qquad A\cup\emptyset = A, \qquad\mbox{and}\qquad A-\emptyset = A.\] From the definition of set difference, we find \(\emptyset-A = \emptyset\). Finally, \(\overline{\overline{A}} = A\).

Example \(\PageIndex{4}\label{eg:unionint-04}\)

Write, in interval notation, \([5,8)\cup(6,9]\) and \([5,8)\cap(6,9]\).

The answers are \[[5,8)\cup(6,9] = [5,9], \qquad\mbox{and}\qquad [5,8)\cap(6,9] = (6,8).\] They are obtained by comparing the location of the two intervals on the real number line.

hands-on exercise \(\PageIndex{4}\label{he:unionint-04}\)

Write, in interval notation, \((0,3)\cup[-1,2)\) and \((0,3)\cap[-1,2)\).

Example \(\PageIndex{5}\label{eg:unionint-05}\)

We are now able to describe the following set \[\{x\in\mathbb{R} \mid (x<5) \vee (x>7)\}\] in the interval notation. It can be written as either \((-\infty,5)\cup(7,\infty)\) or, using complement, \(\mathbb{R}-[5,7\,]\). Consequently, saying \(x\notin[5,7\,]\) is the same as saying \(x\in(-\infty,5) \cup(7,\infty)\), or equivalently, \(x\in \mathbb{R}-[5,7\,]\).

## To Prove a Set is Empty

To prove a set is empty , use a proof by contradiction with these steps:

(1) Assume not. That, is assume \(\ldots\) is not empty.

(2) This means there is an element is \(\ldots\) by definition of the empty set.

(3) Let \(x \in \ldots \).

(4) Come to a contradition and wrap up the proof.

Example \(\PageIndex{6}\)

Prove: \(\forall A \in {\cal U}, A \cap \emptyset = \emptyset.\)

Proof: Assume not. That is, assume for some set \(A,\) \(A \cap \emptyset \neq \emptyset.\) By definition of the empty set, this means there is an element in \(A \cap \emptyset .\)

Let \(x \in A \cap \emptyset .\)

\(x \in A \wedge x\in \emptyset\) by definition of intersection.

This says \(x \in \emptyset \), but the empty set has no elements! This is a contradiction!

Thus, our assumption is false, and the original statement is true. \(\forall A \in {\cal U}, A \cap \emptyset = \emptyset.\)

## Set Properties

(a) These properties should make sense to you and you should be able to prove them. However, you are not to use them as reasons in a proof. Rather your justifications for steps in a proof need to come directly from definitions. The exception to this is DeMorgan's Laws which you may reference as a reason in a proof.

(b) You do not need to memorize these properties or their names. However, you should know the meanings of: commutative, associative and distributive. Also, you should know DeMorgan's Laws by name and substance.

The following properties hold for any sets \(A\), \(B\), and \(C\) in a universal set \({\cal U}\).

- Commutative properties : \(\begin{array}[t]{l} A \cup B = B \cup A, \\ A \cap B = B \cap A. \end{array}\)
- Associative properties : \(\begin{array}[t]{l} (A \cup B) \cup C = A \cup (B \cup C), \\ (A \cap B) \cap C = A \cap (B \cap C). \end{array}\)
- Distributive laws : \(\begin{array}[t]{l} A \cup (B \cap C) = (A \cup B) \cap (A \cup C), \\ A \cap (B \cup C) = (A \cap B) \cup (A \cap C). \end{array}\)
- Idempotent laws : \(\begin{array}[t]{l} A \cup A = A, \\ A \cap A = A. \end{array}\)
- De Morgan’s laws : \(\begin{array}[t]{l} \mbox{ (a) } \overline{A \cup B} = \overline{A} \cap \overline{B}, \\ \mbox{ (b) } \overline{A \cap B} = \overline{A} \cup \overline{B}. \end{array}\)
- Laws of the excluded middle , or inverse laws : \(\begin{array}[t]{l} A \cup \overline{A} = {\cal U}, \\ A \cap \overline{A} = \emptyset. \end{array}\)

As an illustration, we shall prove the distributive law \[A \cup (B \cap C) = (A \cup B) \cap (A \cup C).\]

We need to show that \[A \cup (B \cap C) \subseteq (A \cup B) \cap (A \cup C), \qquad\mbox{and}\qquad (A \cup B) \cap (A \cup C) \subseteq A \cup (B \cap C).\]

Here is a proof of the distributive law \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\).

hands-on exercise \(\PageIndex{5}\label{he:unionint-05}\)

Prove that \(A\cap(B\cup C) = (A\cap B)\cup(A\cap C)\).

hands-on exercise \(\PageIndex{6}\label{he:unionint-06}\)

Prove that if \(A\subseteq B\) and \(A\subseteq C\), then \(A\subseteq B\cap C\).

Let us start with a draft. The statement we want to prove takes the form of \[(A\subseteq B) \wedge (A\subseteq C) \Rightarrow A\subseteq B\cap C.\] Hence, what do we assume and what do we want to prove?

Did you put down we assume \(A\subseteq B\) and \(A\subseteq C\), and we want to prove \(A\subseteq B\cap C\)? Great! Now, what does it mean by \(A\subseteq B\)? How about \(A\subseteq C\)? What is the meaning of \(A\subseteq B\cap C\)?

How can you use the first two pieces of information to obtain what we need to establish?

Now it is time to put everything together, and polish it into a final version. Remember three things:

- the outline of the proof,
- the reason in each step of the main argument, and
- the introduction and the conclusion.

Put the complete proof in the space below.

Here are two results involving complements.

Theorem \(\PageIndex{1}\label{thm:subsetsbar}\)

For any two sets \(A\) and \(B\), we have \(A \subseteq B \Leftrightarrow \overline{B} \subseteq \overline{A}\).

Theorem \(\PageIndex{2}\label{thm:genDeMor}\)

For any sets \(A\), \(B\) and \(C\),

(a) \(A-(B \cup C)=(A-B) \cap (A-C)\)

(b) \(A-(B \cap C)=(A-B) \cup (A-C)\)

## Summary and Review

- Memorize the definitions of intersection, union, and set difference. We rely on them to prove or derive new results.
- The intersection of two sets \(A\) and \(B\), denoted \(A\cap B\), is the set of elements common to both \(A\) and \(B\). In symbols, \(\forall x\in{\cal U}\,\big[x\in A\cap B \Leftrightarrow (x\in A \wedge x\in B)\big]\).
- The union of two sets \(A\) and \(B\), denoted \(A\cup B\), is the set that combines all the elements in \(A\) and \(B\). In symbols, \(\forall x\in{\cal U}\,\big[x\in A\cup B \Leftrightarrow (x\in A\vee x\in B)\big]\).
- The set difference between two sets \(A\) and \(B\), denoted by \(A-B\), is the set of elements that can only be found in \(A\) but not in \(B\). In symbols, it means \(\forall x\in{\cal U}\, \big[x\in A-B \Leftrightarrow (x\in A \wedge x\notin B)\big]\).
- The symmetric difference between two sets \(A\) and \(B\), denoted by \(A \bigtriangleup B\), is the set of elements that can be found in \(A\) and in \(B\), but not in both \(A\) and \(B\). In symbols, it means \(\forall x\in{\cal U}\, \big[x\in A \bigtriangleup B \Leftrightarrow x\in A-B \vee x\in B-A)\big]\).

## Exercises

Exercise \(\PageIndex{1}\label{ex:unionint-01}\)

Write each of the following sets by listing its elements explicitly.

(a) \([-4,4]\cap\mathbb{Z}\)

(b) \((-4,4]\cap\mathbb{Z}\)

(c) \((-4,\infty)\cap\mathbb{Z}\)

(d) \((-\infty,4]\cap\mathbb{N}\)

(e) \((-4,\infty)\cap\mathbb{Z}^-\)

(f) \((4,5)\cap\mathbb{Z}\)

(a) \(\{-4,-3,-2,-1,0,1,2,3,4\}\)

(b) \(\{-3,-2,-1,0,1,2,3,4\}\)

(c) \(\{-3,-2,-1,0,1,2,3,\ldots\}\)

Exercise \(\PageIndex{2}\label{ex:unionint-02}\)

Assume \({\cal U} = \mathbb{Z}\), and let

\(A=\{\ldots, -6,-4,-2,0,2,4,6, \ldots \} = 2\mathbb{Z},\)

\(B=\{\ldots, -9,-6,-3,0,3,6,9, \ldots \} = 3\mathbb{Z},\)

\(C=\{\ldots, -12,-8,-4,0,4,8,12, \ldots \} = 4\mathbb{Z}.\)

Describe the following sets by listing their elements explicitly.

(a) \(A\cap B\)

(b) \(C-A\)

(c) \(A-B\)

(d) \(A\cap\overline{B}\)

(e) \(B-A\)

(f) \(B\cup C\)

(g) \((A\cup B)\cap C\)

(h) \((A\cup B)-C\)

Exercise \(\PageIndex{3}\label{ex:unionint-03}\)

Are these statements true or false?

(a) \([1,2]\cap[2,3] = \emptyset\)

(b) \([1,2)\cup(2,3] = [2,3]\)

(a) false (b) false

Exercise \(\PageIndex{4}\label{ex:unionint-04}\)

Let the universal set \({\cal U}\) be the set of people who voted in the 2012 U.S. presidential election. Define the subsets \(D\), \(B\), and \(W\) of \({\cal U}\) as follows: \[\begin{aligned} D &=& \{x\in{\cal U} \mid x \mbox{ registered as a Democrat}\}, \\ B &=& \{x\in{\cal U} \mid x \mbox{ voted for Barack Obama}\}, \\ W &=& \{x\in{\cal U} \mid x \mbox{ belonged to a union}\}. \end{aligned}\] Express the following subsets of \({\cal U}\) in terms of \(D\), \(B\), and \(W\).

(a) People who did not vote for Barack Obama.

(b) Union members who voted for Barack Obama.

(c) Registered Democrats who voted for Barack Obama but did not belong to a union.

(d) Union members who either were not registered as Democrats or voted for Barack Obama.

(e) People who voted for Barack Obama but were not registered as Democrats and were not union members.

(f) People who were either registered as Democrats and were union members, or did not vote for Barack Obama.

Exercise \(\PageIndex{5}\label{ex:unionint-05}\)

An insurance company classifies its set \({\cal U}\) of policy holders by the following sets: \[\begin{aligned} A &=& \{x\mid x\mbox{ drives a subcompact car}\}, \\ B &=& \{x\mid x\mbox{ drives a car older than 5 years}\}, \\ C &=& \{x\mid x\mbox{ is married}\}, \\ D &=& \{x\mid x\mbox{ is over 21 years old}\}, \\ E &=& \{x\mid x\mbox{ is a male}\}. \end{aligned}\] Describe each of the following subsets of \({\cal U}\) in terms of \(A\), \(B\), \(C\), \(D\), and \(E\).

(a) Male policy holders over 21 years old.

(b) Policy holders who are either female or drive cars more than 5 years old.

(c) Female policy holders over 21 years old who drive subcompact cars.

(d) Male policy holders who are either married or over 21 years old and do not drive subcompact cars.

(a) \(E\cap D\) (b) \(\overline{E}\cup B\)

Exercise \(\PageIndex{6}\label{ex:unionint-06}\)

Let \(A\) and \(B\) be arbitrary sets. Complete the following statements.

(a) \(A\subseteq B \Leftrightarrow A\cap B = \) ___________________

(b) \(A\subseteq B \Leftrightarrow A\cup B = \) ___________________

(c) \(A\subseteq B \Leftrightarrow A - B = \) ___________________

(d) \(A\subset B \Leftrightarrow (A-B= \) ___________________\(\wedge\,B-A\neq\) ___________________ \()\)

(e) \(A\subset B \Leftrightarrow (A\cap B=\) ___________________\(\wedge\,A\cap B\neq\) ___________________ \()\)

(f) \(A - B = B - A \Leftrightarrow \) ___________________

Exercise \(\PageIndex{7}\label{ex:unionint-07}\)

Give examples of sets \(A\) and \(B\) such that \(A\in B\) and \(A\subset B\).

For example, take \(A=\{x\}\), and \(B=\{\{x\},x\}\).

Exercise \(\PageIndex{8}\label{ex:unionint-08}\)

(a) Prove De Morgan’s law, (a) .

(b) Prove De Morgan’s law, (b) .

Exercise \(\PageIndex{9}\label{ex:unionint-09}\)

Let \(A\), \(B\), and \(C\) be any three sets. Prove that if \(A\subseteq C\) and \(B\subseteq C\), then \(A\cup B\subseteq C\).

Assume \(A\subseteq C\) and \(B\subseteq C\), we want to show that \(A\cup B \subseteq C\).

Let \(x\in A\cup B\). we want to show that \(x\in C\) as well.

Since \(x\in A\cup B\), then either \(x\in A\) or \(x\in B\) by definition of union.

Case 1: If \(x\in A\), then \(A\subseteq C\) implies that \(x\in C\) by definition of subset.

Case 2: If \(x\in B\), then \(B\subseteq C\) implies that \(x\in C\) by definition of subset.

In both cases, we find \(x\in C\). So, if \(x\in A\cup B\) then \(x\in C\).

This proves that \(A\cup B\subseteq C\) by definition of subset.

\(\therefore\) For any sets \(A\), \(B\), and \(C\) if \(A\subseteq C\) and \(B\subseteq C\), then \(A\cup B\subseteq C\).

Exercise \(\PageIndex{10}\label{ex:unionint-10}\)

Prove Theorem 4.3.1

Exercise \(\PageIndex{11}\label{ex:unionint-11}\)

(a) Prove Theorem 4.3.2 part (a)

(b) Prove Theorem 4.3.2 part (b)

Exercise \(\PageIndex{12}\label{ex:unionint-12}\)

Let \(A\), \(B\), and \(C\) be any three sets. Prove that

(a) \(A-B=A\cap\overline{B}\)

(b) \(A=(A-B)\cup(A\cap B)\)

(c) \(A-(B-C) = A\cap(\overline{B}\cup C)\)

(d) \((A-B)-C = A-(B\cup C)\)

Exercise \(\PageIndex{13}\label{ex:unionint-13}\)

Comment on the following statements. Are they syntactically correct?

(a) \(x\in A \cap x\in B \equiv x\in A\cap B\)

(b) \(x\in A\wedge B \Rightarrow x\in A\cap B\)

(a) The notation \(\cap\) is used to connect two sets, but “\(x\in A\)” and “\(x\in B\)” are both logical statements. We should also use \(\Leftrightarrow\) instead of \(\equiv\). The statement should have been written as “\(x\in A \,\wedge\, x\in B \Leftrightarrow x\in A\cap B\).”

(b) If we read it aloud, it sounds perfect: \[\mbox{If $x$ belongs to $A$ and $B$, then $x$ belongs to $A\cap B$}.\] The trouble is, every notation has its own meaning and specific usage. In this case, \(\wedge\) is not exactly a replacement for the English word “and.” Instead, it is the notation for joining two logical statements to form a conjunction. Before \(\wedge\), we have “\(x\in A\),” which is a logical statement. But, after \(\wedge\), we have “\(B\),” which is a set, and not a logical statement. It should be written as “\(x\in A\,\wedge\,x\in B \Rightarrow x\in A\cap B\).”

Exercise \(\PageIndex{14}\label{ex:unionint-14}\)

Prove or disprove each of the following statements about arbitrary sets \(A\) and \(B\). If you think a statement is true, prove it; if you think it is false, provide a counterexample.

(a) \(\mathscr{P}(A\cap B) = \mathscr{P}(A)\cap\mathscr{P}(B)\)

(b) \(\mathscr{P}(A\cup B) = \mathscr{P}(A)\cup\mathscr{P}(B)\)

(c) \(\mathscr{P}(A - B) = \mathscr{P}(A) - \mathscr{P}(B)\)

To show that two sets \(U\) and \(V\) are equal, we usually want to prove that \(U \subseteq V\) and \(V \subseteq U\). For the subset relationship, we start with let \(x\in U \). In this problem, the element \(x\) is actually a set. Since we usually use uppercase letters to denote sets, for (a) we should start the proof of the subset relationship “Let \(S\in\mathscr{P}(A\cap B)\),” using an uppercase letter to emphasize the elements of \(\mathscr{P}(A\cap B)\) are sets. These remarks also apply to (b) and (c).

Exercise \(\PageIndex{15}\)

Let \({\cal U}=\{1,2,3,4,5,6,7,8\}\), \(A=\{2,4,6,8\}\), \(B=\{3,5\}\), \(C=\{1,2,3,4\}\) and\(D=\{6,8\}\). Find

(a) \(A\cap C\) (b) \(A\cap B\) (c) \(\emptyset \cup B\)

(d) \(\emptyset \cap B\) (e) \(A-(B \cup C)\) (f) \(C-B\)

(g) \(A\bigtriangleup C\) (h) \(A \cup {\cal U}\) (i) \(A\cap D\)

(j) \(A\cup D\) (k) \(B\cap D\) (l) \(B\bigtriangleup C\)

(m) \(A \cap {\cal U}\) (n) \(\overline{A}\) (o) \(\overline{B}\).

(p) \(D \cup (B \cap C)\) (q) \(\overline{A \cup C}\) (r) \(\overline{A} \cup \overline{C} \)

(s) Which pairs of sets are disjoint?

(a) \(\{2,4\}\) (b) \(\emptyset \) (c) \(B\) (d) \(\emptyset\)

Exercise \(\PageIndex{16}\)

If \(A \subseteq B\) then \(A-B= \emptyset.\)

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## Statistics and probability

Course: statistics and probability > unit 7, intersection and union of sets.

- Relative complement or difference between sets
- Universal set and absolute complement
- Subset, strict subset, and superset
- Bringing the set operations together
- Basic set notation

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## Video transcript

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## Union and Intersection

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The union of 2 sets \(A\) and \(B\) is denoted by \( A \cup B \). This is the set of all distinct elements that are in \(A\) or \(B \). A useful way to remember the symbol is \(\cup\)nion. We can define the union of a collection of sets, as the set of all distinct elements that are in any of these sets.

The intersection of 2 sets \(A\) and \(B\) is denoted by \(A \cap B \). This is the set of all distinct elements that are in both \(A\) and \(B\). A useful way to remember the symbol is i\(\cap\)tersection. We define the intersection of a collection of sets, as the set of all distinct elements that are in all of these sets.

If \( A = \{ 1, 3, 5, 7, 9 \} \) and \( B = \{ 2, 3, 5, 7, \} \), what are \( A \cup B \) and \( A \cap B \)? We have \[\begin{align} A \cup B &= \{ 1, 2, 3, 5, 7, 9 \} \\ A \cap B &= \{ 3, 5, 7 \}. \ _\square \end{align}\]

A great way of thinking about union and intersection is by using Venn diagrams. These are explained as follows:

We will represent sets with circles.

Then we can put the values in appropriate areas.

The Union is any region including either A or B.

The Intersection is any region including both A and B.

The diagrams we have drawn are called the Venn diagrams.

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## The Intersection & Union of Sets

Oct 01, 2014

150 likes | 340 Views

The Intersection & Union of Sets. Algebra I Mrs. Stoltzfus Lower Dauphin School District. Consider these Sets. Actors. Musicians. Miley Cyrus Beyonce Gwyneth Paltrow Eminem. Intersection.

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## Presentation Transcript

The Intersection & Union of Sets Algebra I Mrs. Stoltzfus Lower Dauphin School District

Consider these Sets Actors Musicians Miley Cyrus Beyonce Gwyneth Paltrow Eminem Intersection Musicians ∪ Actors = {Lady Gaga, Coldplay, John Mayer, Miley Cyrus, Beyonce, Gwyneth Paltrow, Eminem, Bruce Willis, Julia Roberts, Matt Damon} Musicians Actors = { Miley Cyrus, Beyonce, Gwyneth Paltrow, Eminem}

Vocabulary/Notation

Vocabulary/Notation • Disjoint Sets have no element in common

Example A Q R 6 7 2 4 1 5 • The numbered shapes below are to be sorted into two sets, R and Q. • Set R contains shapes with a right-angle (90°). • Set Q contains shapes with four sides.

= {3, 7, 11} B = {0,1,3,4,5,7,8,9,11,13} = {11} Example B A C 3 7 0 11 4 = {0,2,3,4,7,11} Express {0, 4, 11} in terms of A, B, and C Terms of A, B, and C.

R = {the real numbers between -10 and -3} Example C ={all real numbers less than or equal to -2} B = {the real numbers less than or equal to -2} ={all real numbers between -10 and -3}

Homework: • Page 476-477, #1-26

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## 7.2 Union, intersection, complement of an event, odds

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## Union, Intersection, and Compound Inequalities

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## The Intersection of Finance and Insurance

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## Objective- To use intersection and union to simplify problems involving sets.

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## Union of the Comoros

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## Union, Intersection, Difference

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## ”State of the Union”

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## 3.8 Unions and Intersections of Sets. Set operations Let U = {x|x is an English-language film} Set A below contains the five best films according to the.

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## Presentation on theme: "3.8 Unions and Intersections of Sets. Set operations Let U = {x|x is an English-language film} Set A below contains the five best films according to the."— Presentation transcript:

Introduction to Set Theory

Set Operations and Venn Diagrams 2.2 – 2.3. The intersection of sets A and B, denoted by, is the set of all elements that are common to both. That is,.

Part 1 Module 2 Set operations, Venn diagrams

1. Number of Elements in A 2. Inclusion-Exclusion Principle 3. Venn Diagram 4. De Morgan's Laws 1.

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Chapter 3: Set Theory and Logic

Venn Diagrams/Set Theory Venn Diagram- A picture that illustrates the relationships between two or more sets { } are often used to denote members of.

SOLUTION EXAMPLE 4 Identify angle pairs To find vertical angles, look or angles formed by intersecting lines. To find linear pairs, look for adjacent angles.

VENN DIAGRAMS. Venn Diagrams A Venn diagram is a drawing in which sets are represented by geometric figures such as circles and rectangles. Venn diagrams.

Problems of the Day Solve each inequality. 1. x + 3 ≤ ≥ 3x x + 1 ≤ 25 x ≤ 7 23 < –2x + 3 x ˂ –10 Solve each inequality and graph the.

THE NATURE OF SETS Copyright © Cengage Learning. All rights reserved. 2.

© 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 2 Set Theory.

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Section 2.3 Using Venn Diagrams to Study Set Operations Math in Our World.

Set Operations Chapter 2 Sec 3. Union What does the word mean to you? What does it mean in mathematics?

Objectives: Find the union and intersection of sets. Count the elements of sets. Apply the Addition of Probabilities Principle. Standard Addressed:

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## How to Merge Shapes in PowerPoint (Union, Intersect, Subtract)

Group shapes is a functionality in PowerPoint that helps to merge shapes, via set operations such as Union, Intersect, Substract and more. With Group Shapes, you can easily combine multiple shapes into a single grouped entity. This can be helpful when you need to create complex shapes or when you need to combine shapes with different properties.

This is a fantastic tool for presentation designers as it helps to creative diagrams and graphics in PowerPoint that are very easy to edit or resize without losing image quality. Moreover, you can also apply specific shape properties to the grouped entity, such as background color, border styles, etc.

The Merge Shapes functionality is a pro tool that you can find under the Shape Format menu in the PowerPoint Ribbon.

To access the Merge Shapes feature in Microsoft PowerPoint, select the shapes you want to group or apply the set operation, and then choose the Shape Format menu. Click on Merge Shapes button to display the popup menu with the available operations.

To use the Combine Shapes or Merge Shapes feature, you’d need to select the shapes you want to apply set operations. The available functions are Union, Combine, Fragment, Intersect and Subtract.

The picture above shows the Merge Shapes menu in PowerPoint, and we are showing how to apply the operations to the free Circular Diagram template for PowerPoint with 3 elements that you can download for free.

Alternatively you can add the tools to the Quick Access Toolbar. This can help you to access the features more quickly. In order to configure the toolbar you can click in the small toolbar arrow and then click Customize Quick Access Toolbar.

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Union & Intersection of Sets - Download as a PDF or view online for free. Submit Search. Upload. Union & Intersection of Sets ... Grade 7 Sets.ppt. Grade 7 Sets.ppt ...

24. Learning Task 2 Find the union and intersection of events A and B defined by the following situations. 1. A four-section spinner marked red, yellow, blue, and black is spun once and a five peso coin is tossed once. A is the event of spinning a yellow and B is the event of getting a head. 2.

Ch 3-8 Students will be able to find the unions and intersections of sets. Certain regions of the Venn diagram in the Solve It show unions and intersections of sets. Focus Question What are the union and intersection of sets? The union of two or more sets is the set that contains all of the elements of the sets. The intersection of two or more sets is the set of elements that are common to ...

Ä - 2 z ó Ÿ ¨ Subsets¡ Ÿ ¨ "A is a subset of B if every element of A is an element of B. Notation: For each set A, For each set B, A is proper subset of B if and ¡ È• ó Ÿ ¨ Unions¡ Ÿ ~The union of two sets A and B is The word or is inclusive.¡

Demo -Union of Two Events. 4. Intersection of Two Events. The intersection of two events A and B is the. event that occurs if both A and B occur on a. single performance of the experiment. We write. AnB for the intersection of a and B. AnB. consists of all sample points that belong to both.

Illustrate 𝑃 ∩ 𝑄 using Venn diagram. 17. UNION AND INTERSECTION OF SETS The union of two sets are all the elements from both sets. Thus, the union of sets A and B, written as 𝐴 ∪ 𝐵, is the set of the elements that are members of A, or members of B ,or members of both A and B. 18.

Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds. Maths sets ppt. Probability - Independent & Dependent Events. 4.4 probability of compound events. 12.4 probability of compound events. Ppt sets and set operations. Probability Powerpoint. PROBABILITY. Multiplication and Division of Integers.

We rely on them to prove or derive new results. The intersection of two sets A and B, denoted A ∩ B, is the set of elements common to both A and B. In symbols, ∀x ∈ U [x ∈ A ∩ B ⇔ (x ∈ A ∧ x ∈ B)]. The union of two sets A and B, denoted A ∪ B, is the set that combines all the elements in A and B.

Presentation Transcript. 3-8 Unions and Intersection of Sets. The Union of two or more sets is the set that contains all elements of the sets Symbol: How to find: list the elements that are in either set, or in both sets. Problem 1: Union of Sets In your left pocket, you have a quarter, a paper clip, and a key.

So I'll put a 12 here. The number 7 is only in A. And the number, I guess, 13, 10 and 3 is only in B, so we're done. The set of 4 and 12 is the intersection of sets A and B. And we could even, if we want to, we could even label this as a new set. We could say set C is the intersection of A and B, and it's this set right over here.

3-8 Unions and Intersections of Sets Objective: To find union and intersection of sets.. 3-8 Unions and Intersections of Sets • On page 214 Solve the Get Ready! • www.pearsonsuccessnet.com. 3-8 Unions and Intersections of Sets • The union of two or more sets is the set that contains ALL elements of the sets. • The symbol for union is • To find the union of two sets.

Union and Intersection. The union of 2 sets A A and B B is denoted by A \cup B A∪ B. This is the set of all distinct elements that are in A A or B B. A useful way to remember the symbol is \cup ∪ nion. We can define the union of a collection of sets, as the set of all distinct elements that are in any of these sets.

Presentation on theme: "Intersection and Union of Sets"— Presentation transcript: 1 Intersection and Union of Sets 2 Objectives Understand the difference between intersection and union of sets.

8 Find the union ( ) and intersection ( ) of each pair of sets. 3) U = {whole numbers from 1-20 inclusive} P = {prime numbers less than 20} D = {odd whole numbers less than 20} 4) U = {letters of the English alphabet} F = {different letters in the word 'foot'} B = {different letters in the word 'base'} Note : sets F and B are disjoint!

7.2 Union, intersection, complement of an event, odds. 7.2 Union, intersection, complement of an event, odds. In this section, we will develop the rules of probability for compound events ( more than one event) and will discuss probabilities involving the union of events as well as intersection of two events. 593 views • 11 slides

Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/statistics-probability/probabi...

the union of sets A and B, written as AUB , is the set of. the elements that are members of A,or members of B ,or members of both A and B. Union of Sets "U" AUB Union of Sets The union of two sets is everything that is contained within the two circles joined together. It is the combined total of the two sets, where each item is only listed ...

Download ppt "3.8 Unions and Intersections of Sets. Set operations Let U = {x|x is an English-language film} Set A below contains the five best films according to the." ... Similar presentations . Introduction to Set Theory. Set Operations and Venn Diagrams 2.2 - 2.3. The intersection of sets A and B, denoted by, is the set of all elements ...

Union and Intersection of Sets - Free download as Powerpoint Presentation (.ppt), PDF File (.pdf), Text File (.txt) or view presentation slides online. Scribd is the world's largest social reading and publishing site.

Ppt sets and set operations geckbanaag. Union & Intersection of Sets myla gambalan. Operations on Sets Free Math Powerpoints. 1. sets and basic notations eduardman. Introduction to Sets Ashita Agrawal. Operations on sets myra concepcion.

This contains all of the elements which are in both A and B. 6. Union and Intersection with empty. 7. Union and Intersection in Psql. 8. Union - A hot startup is holding a special event and wants to send out invites for my marriage ( ) to some of my best clients and also to some VIPs.

To access the Merge Shapes feature in Microsoft PowerPoint, select the shapes you want to group or apply the set operation, and then choose the Shape Format menu. Click on Merge Shapes button to display the popup menu with the available operations. To use the Combine Shapes or Merge Shapes feature, you'd need to select the shapes you want to ...

3. Learn about it! • A Venn diagram is used to represent relationships between a collection of objects or sets. • As shown below, a Venn diagram usually has a rectangle with circles inside. The rectangle represents the universal set and the circles inside represent the different subsets in the universal set.