Section 1.1 Sets

Sets A set is a well-defined collection of objects.

Elements The objects inside of a set are called elements of the set.

a 2 A ( a is an element of A )

a /2 A ( a is not an element of A)

Roster and Set-Builder Notation

Roster notation will be used most commonly in this class, and consists of listing the elements of a set

in between curly braces. Set-builder notation is when a rule is used to define a definite property that

an object must have in order to be in the set.

1. Let A be the set of all letters in the English alphabet.

(a) Write A in roster notation and in set-builder notation.

(b) Is the greek letter � an element of A?

Set Equality Two sets A and B are equal, written A = B, if and only if they have exactly the

same elements.

2. Let A = {a, e, l, t, r}. Which of the following sets are equal to A? (Choose all that apply.)

(a) {x | x is a letter of the word latter}

(b) {x | x is a letter of the word later}

(c) {x | x is a letter of the word late}

(d) {x | x is a letter of the word rated}

(e) {x | x is a letter of the word relate}

Subset If every element of a set A is also an element of a set B, then we say that A is a subset

of B and we write A ✓ B.

Note: If we write A ⇢ B, then this means that A is a proper subset of B, without the possibility

of equality. Therefore, for any set A, A is NOT a proper subset of itself.

3. If A = {u, v, y, z} and B = {x, y, z}, determine whether the following statements are true or false.

(a) x, y 2 B

(b) {x, y, z} ⇢ B

(c) {u, w} /2 A

(d) {x, w} ✓ A

The Empty and Universal Set The set that contains no elements is called the empty set

and the symbol for the empty set is ?. The set of all elements under discussion is called the

universal set and is usually denoted by U .

Note: The empty set is a subset of every set. That is, ? ✓ A where A is any set.

Set Operations

Set Union Let A and B be sets. The union of A and B, written A[B, is the set of all elements

that belong to either A or B or both. This is like adding the two sets. Below is a Venn Diagram

illustrating the set A [ B.

A B

A [ B

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Set Intersection Let A and B be sets. The intersection of A and B, written A\B, is the set

of all elements that belong to both A and B. This is what the two sets have in common. Below

is a venn diagram illustrating the set A \ B.

A B

A \ B

Complement of a Set If U is a universal set and A is a subset of U , then the set of all elements

in U that are not in A is called the complement of A and is denoted A

c. Below are venn diagrams

illustrating the sets Ac and B

c.

A

cB

c

4. If A and B are two subsets of a universal set U , illustrate the sets Ac \B and A\B

c using venn

diagrams.

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Set Complementation

If U is a universal set and A is a subset of U , then

a. U c = ? b. ?c = U c. (Ac)c = A

d. A [ A

c = U e. A \ A

c = ?

Properties of Set Operations

Let U be a universal set. If A, B, and C are arbitrary subsets of U , then

A [ B = B [ A Commutative law for union

A \ B = B \ A Commutative law for intersection

A [ (B [ C) = (A [B) [ C Associative law for union

A \ (B \ C) = (A \B) \ C Associative law for intersection

A [ (B \ C) = (A [ B) \ (A [ C) Distributive law for union

A \ (B [ C) = (A \ B) [ (A \ C) Distributive law for intersection

De Morgan’s Laws

Let A and B be sets. Then

(A [ B)c = A

c \B

c

(A \ B)c = A

c [B

c

5. Write venn diagrams to represent each of the following sets.

(a) A [ B

c

(b) A

c \ B

c

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g@GjGEonbsn.eregion .

.

6. Write venn diagrams to represent each of the following sets.

(a) A \ B \ C

c

(b) A

c [ B [ C

Disjoint Sets Two sets A and B are disjoint if and only if they have no elements in common.

That is, if A \ B = ?.

7. Let the universal set U = {a, b, c, d} with sets A = {a, b} and B = {b, c}. Find the set A [ B

c.

Which of the sets below are disjoint to this set?

(a) A

c \ B

(b) A

c \ B

c

(c) A

c [ B

c

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u§€§

Onlyshade region b.

8. Let U denote the set of all senators in Congress and let

D = {x is in U | x is a Democrat}

R = {x is in U | x is a Republican}

F = {x is in U | x is a female}

L = {x is in U | x is a lawyer}.

Write the set that represents each statement.

(a) The set of all Republicans who are female or are lawyers.

(b) The set of all senators who are not Republicans or are lawyers

Are the sets in parts (a) and (b) disjoint?

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knifeNo

. They have republican lawyers in common .

Even though part (b) has Rc , the union allows

for all lawyers to be included, even republican lawyers .

9. Let U = {-8, -6, -3, 1, 3, 8, 11, 14, 16, 20}, A = {-8, -3, 3, 11, 16}, B = {-6, 1, 8, 14, 20}, andC = {-8, -6, 1, 3, 14, 16}. Find each set using roster notation.

(a) A

c \ (B \ C

c)

(b) (A [ B)c \ C

c

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'- Acn{ 8,20 }

Bnc '= 58,203

Atf -6, 1,8 , 14,203

4- 6,48114120 }n{ 8,20 }

={8@=AcnBcn(

cDemozan 's

=µtAuB5=AcnBe

No elements are outside all three sets.