Example

Some examples of sets defined by describing the contents:

1. [latex]\text{C}\;=\;\text{the set of all books written about travel to Chile}[/latex]. This is a finite set.
2. [latex]\mathbb{Z}=\text{the set of all integers}[/latex]. This is an infinite set.

Some examples of sets defined by listing the elements of the set:

1. [latex]A\;= \;\{1, 3, 9, 12\}[/latex].This is a finite set.
2. [latex]\text{R}\;=\;\{\text{red, orange, yellow, green, blue, indigo, purple}\}[/latex]. This is a finite set.

Example

Let [latex]A\;= \;\{1, 2, 3, 4\}[/latex]

To say that 2 is an element of the set, we write [latex]2\;\in\;A[/latex].

Example

Consider these three sets:

[latex]A=\text{the set of all even numbers}[/latex]
[latex]B=\{2,4,6\}[/latex]
[latex]C=\{2,3,4,6\}[/latex]

Here [latex]B\:\subset\:A[/latex] since every element of [latex]B[/latex] is an even number, so is an element of [latex]A[/latex].

More formally, we could say [latex]B\:\subset\:A[/latex] since if [latex]x\:\in\:B[/latex], then [latex]x\:\in\:A[/latex].

It is also true that [latex]B\:\subset\:C[/latex] because every element of [latex]B[/latex] is an element of [latex]C[/latex].

[latex]C[/latex] is NOT a subset of [latex]A[/latex], since [latex]C[/latex] contains an element, 3, that is not contained in [latex]A[/latex].

Example

Suppose a set contains the plays “Much Ado About Nothing,” “MacBeth,” and “A Midsummer’s Night Dream.” What is a larger set this might be a subset of?

There are many possible answers here. One would be the set of plays by Shakespeare. This is also a subset of the set of all plays ever written. It is also a subset of all British literature.

Example

1. If we are discussing searching for books, the universal set might be all the books in the library.
2. If we are grouping our Facebook friends, the universal set would be all of our Facebook friends.
3. If we are working with sets of numbers, the universal set might be all whole numbers, all integers, or all real numbers.

Example

Suppose the universal set is [latex]U[/latex] = all whole numbers from 1 to 9. i.e. [latex]U=\{1,2,3,4,5,6,7,8,9\}[/latex]

If [latex]A=\{1, 2, 4\}[/latex], then [latex]A^{\prime}=\{3,5,6,7,8,9\}[/latex].

Example

Consider the sets:

[latex]A=\{\text{red, green, blue}\}[/latex]
[latex]B=\{\text{red, yellow, orange}\}[/latex]
[latex]C=\{\text{red, orange, yellow, green, blue, purple}\}[/latex]

Find the following:

1. Find [latex]A\cup\,B[/latex]
2. Find [latex]A\cap\,B[/latex]
3. Find [latex]A^{\prime}\cap\,C[/latex]

Answers

1. The union contains all the elements in either set: [latex]A\cup\,B=\{\text{red, green, blue, yellow, orange}\}[/latex]. Notice we only list red once.
2. The intersection contains all the elements in both sets: [latex]A\cap\,B=\{\text{red}\}[/latex]. Red is the only color in both sets.
3. Here we’re looking for all the elements that are not in set [latex]A[/latex] but are in set [latex]C[/latex]:  [latex]A^{\prime}\cap\,C=\{\text{orange, yellow, purple}\}[/latex]

Example

Suppose [latex]H=\{\text{cat, dog, rabbit, mouse}\}[/latex], [latex]F=\{\text{dog, cow, duck, pig, rabbit}\}[/latex], and [latex]W=\{\text{duck, rabbit, deer, frog, mouse}\}[/latex].

1. Find [latex](H\cap\,F)\cup\,W[/latex]
2. Find [latex]H\cap\,(F\cup\,W)[/latex]
3. Find [latex](H\cap\,F)^{\prime}\cap\,W[/latex]

Solutions

1. We start with the intersection since it is in parentheses: [latex](H\cap\,F)=\{\text{dog, rabbit}\}[/latex]. Now we union that result with [latex]W[/latex]:  [latex](H\cap\,F)\cup\,W=\{\text{dog, duck, rabbit, deer, frog, mouse}\}[/latex].
2. We start with the union inside parentheses: [latex](F\cup\,W)=\{\text{dog, cow, duck, pig, rabbit, deer, frog, mouse}\}[/latex] . Now we intersect that result with [latex]H[/latex]: [latex]H\cap\,(F\cup\,W)=\{\text{dog, rabbit, mouse}\}[/latex].
3. We start with the intersection inside parentheses: [latex](H\cap\,F)=\{\text{dog, rabbit}\}[/latex]. Now we want to find the elements of [latex]W[/latex] that are not in [latex](H\cap\,F)[/latex].  [latex](H\cap\,F)^{\prime}\cap\,W= \{\text{duck, deer, frog, mouse}\}[/latex].

Example

Suppose the universal set is students in a class. [latex]A[/latex] = all students with blonde hair; [latex]B[/latex] = all students with blue eyes; [latex]C[/latex] = all students at least 6 feet tall.

1. What does [latex]B^{\prime}[/latex] represent?

2. What does  [latex]A\cap\,B^{\prime}[/latex] represent?

3. What does  [latex](A\cup C)^{\prime}\cap B[/latex] represent?

Use the Venn to write set interactions to represent each category.

2. Students with blonde hair and blue eyes.

3. Students with blue eyes that are at least 6 feet tall.

4.Students who have blonde hair, blue eyes, and are at least 6 feet tall.

5. Students who are under 6 feet tall and have neither blonde hair nor blue eyes.

6. Students who have neither blonde hair nor blue eyes but are at least 6′ tall.

Example

Create Venn diagrams to illustrate [latex]A\cup\,B[/latex], [latex]A\cap\,B[/latex], and [latex]A^{\prime}\cap\,B[/latex].

[latex]A\cup\,B[/latex] contains all elements in either set.

[latex]A\cap\,B[/latex] contains only those elements in both sets—in the overlap of the circles.

[latex]A^{\prime}[/latex] will contain all elements not in the set [latex]A[/latex]. [latex]A^{\prime}\cap\,B[/latex] will contain the elements in set [latex]B[/latex] that are not in set [latex]A[/latex].

Example

Use a Venn diagram to illustrate [latex](H\cap\,F)^{\prime}\cap\,W[/latex]

We’ll start by identifying everything in the set [latex]H\cap\,F[/latex]

Now, [latex](H\cap\,F)^{\prime}\cap\,W[/latex] will contain everything not in the set identified above that is also in set W.

Example

Create an expression to represent the outlined part of the Venn diagram shown.

The elements in the outlined set are in sets [latex]H[/latex] and [latex]F[/latex], but are not in set [latex]W[/latex]. So we represent this set as [latex]H\cap\,F\cap\,W^{\prime}[/latex].