Section 6.1 Sets and Set Operations

[Pages:14]Section 6.1 Sets and Set Operations

Sets A set is a well-defined collection of objects. The objects in this collection are called elements of

the set. If is an element of the set then we write 2 , if is an element of a set , then we

a

A

a A a not

A

write 2 . a/A

Roster and Set-Builder Notation

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

in between curly braces.

is when a rule is used to define a definite property that

Set-builder notation

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 in roster notation and in set-builder notation. A

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

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

AB

AB

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 is also an element of a set , then we say that is a subset

A

B

A

of and we write .

B

AB

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 = {

} and = { }, determine whether the following statements are true or false.

A u, v, y, z B x, y, z

(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 and be sets. The union of and , written [ , is the set of all elements

AB

AB

AB

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

illustrating the set [ . AB

[ AB

A

B

2 Fall 2017, Maya Johnson

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

AB

AB

AB

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

is a venn diagram illustrating the set \ . AB

A\B

A

B

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

U

A

U

in that are in is called the complement of and is denoted c. Below are venn diagrams

U

not A

A

A

illustrating the sets c and c. AB

c

c

A

B

4. If and are two subsets of a universal set , illustrate the sets c \ and \ c using venn

AB

U

A B AB

diagrams.

3 Fall 2017, Maya Johnson

Set Complementation

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

U

A

U

a.

c

U

=

?

b. ?c = U

c. ( c)c = AA

d. [ c = AA U

e.

\c AA

=

?

Properties of Set Operations

Let be a universal set. If , , and are arbitrary subsets of , then

U

AB C

U

A[B =B[A A\B =B\A A [ (B [ C) = (A [ B) [ C A \ (B \ C) = (A \ B) \ C A [ (B \ C) = (A [ B) \ (A [ C) A \ (B [ C) = (A \ B) [ (A \ C)

Commutative law for union Commutative law for intersection

Associative law for union Associative law for intersection

Distributive law for union Distributive law for intersection

De Morgan's Laws Let A and B be sets. Then

( [ )c = c \ c AB A B ( \ )c = c [ c AB A B

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

(a) [ c AB

(b) c \ c AB

4 Fall 2017, Maya Johnson

6. Write venn diagrams to represent each of the following sets. (a) \ \ c ABC

(b) c [ [ A BC

Disjoint Sets Two sets and are disjoint if and only if they have elements in common.

AB

no

That is, if A \ B = ?.

5 Fall 2017, Maya Johnson

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

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?

6 Fall 2017, Maya Johnson

8. Let = {-9, -6, -1, 2, 5, 7, 11, 13, 17, 19}, = {-9, -1, 5, 11, 17}, = {-6, 2, 7, 13, 19}, and

U

A

B

C

= {-9, -6, 2, 5, 13, 17}. Find each set using roster notation.

(a) ( \ ) [ AB C

(b) ( [ [ )c ABC

(c) ( \ \ )c ABC

-91-6,42/5,711,13174

7 Fall 2017, Maya Johnson

Section 6.2 The Number of Elements in a Finite Set

Counting Problems If a problem requires knowing the number of elements in a given set, then we call such a problem a Counting problem.

Number of Elements in If is a set, then ( ) is the number of elements in the set . If is a

AA

nA

AA

finite set, then we can simply count the number of elements in to find ( ).

A

nA

Note: If is a universal set and is a subset of , then ( c) = ( ) ( )

U

A

U

nA nU nA

1. Let the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Find the following.

(a) ( ) nU

(b) ( c), where = {x | x is an even number from 1 to 10}

nA

A

(c) ( ), where = {1 3 9}

nB

B ,,

(d) n(?)

Addition Rule for Sets: Very Useful Formula

If and are finite sets then AB

n(A [ B) = n(A) + n(B) n(A \ B)

2. If ( ) = 13, ( [ ) = 24, and ( \ ) = 6, find ( ).

nB

nA B

nA B

nA

8 Fall 2017, Maya Johnson

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download