DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.

DP SL StudiesChapter 7

Sets and Venn Diagrams

DP Studies Chapter 7 Homework

Section A: 1, 2, 4, 5, 7, 9

Section B: 2, 4

Section C: 1, 2, 4, 5

Section D: 1, 4, 5, 6

Section E: 1, 4

Section F: 1, 3, 4, 7, 8

Section G: 2, 6, 8

Contents: Sets and Venn diagrams

• A Sets• B Set builder notation• C Complements of sets• D Venn diagrams• E Venn diagram regions• F Numbers in regions• G Problem solving with Venn diagrams

A. Set Notations

A set is a collection of numbers or objects.

Examples:

1. the set of all digits which we use to write numbers is

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

2. set of all vowels, then V = {a, e, i, o, u}.

A. Set Notations

The numbers or objects in a set are called the elements or members of the set.

Examples:

1. So, for the set A = {1, 2, 3, 4, 5, 6, 7} we can say

4 e A (4 is an element of set A), but 9 e A

(9 is not an element of set A).

2. For the set of all vowels, V = {a, e, i, o, u}, we can

say a e V (a is an element of set V), but b e V

(b is not an element of set V).

A. Set Notations

The set { } or 0 is called the empty set and contains no elements.

Example

Let A be the set of all NBA players who are 10 feet tall.

A = {}

A. Set Notations

Special number sets

A. Set Notations

The number of elements in set A is written n(A).

Example:

the set A = {2, 3, 5, 8, 13, 21} has 6 elements, so we write n(A) = 6.

A. Set Notations

A set which has a finite number of elements is called a finite set.

Example:

1. A = {2, 3, 5, 8, 13, 21} is a finite set.

2. Ø is also a finite set, since n(Ø) = 0.

A. Set Notations

Infinite sets are sets which have infinitely many elements.

Example:

1. the set of positive integers {1, 2, 3, 4, ....} does not have a

largest element, but rather keeps on going forever. It is

therefore an infinite set.

2. the sets N , Z , Z+, Z – , Q , and R are all infinite sets.

A. Set Notations

Suppose P and Q are two sets. P is a subset of Q if every element

of P is also an element of Q. We write P Q.

Example:

{2, 3, 5} {1, 2, 3, 4, 5, 6} as every element in the first set is

also in the second set.

We say P is a proper subset of Q if P is a subset of Q but is

not equal to Q.

We write P Q.

A. Set Notations

If P and Q are two sets then

P Q is the intersection of P and Q, and consists of

all elements which are in both P and Q.

P Q is the union of P and Q, and consists of all

elements which are in P or Q.

Examples:

1. If P = {1, 3, 4} and Q = {2, 3, 5} then P Q = {3} and

P Q = {1, 2, 3, 4, 5}

2. The set of integers is made up of the set of negative

integers, zero, and the set of positive integers.

Z = (Z – {0}Z +)

A. Set Notations

Two sets are disjoint or mutually exclusive if they have no elements in common.

Example:

Set A = {0, 2, 4, 6, 8} and Set B = {1, 3, 5, 7}

Set A and Set B are disjoint or mutually exclusive

A. Set Notations

Example 1:

A. Set Notations

Solution to Example 1:

B: Set Builder Notation

Reading a set notation:

A = {x | -2 < x < 4, x e Z}

“the set of all x such that x is an integer between -2 and 4, including -2 and 4.”

We can represent A on a number line as:

A is a finite set, and n(A) = 7.

such thatthe set of all

B: Set Builder Notation

Reading a set notation:

B = {x | -2 < x < 4, x e R}

“the set of all real x such that x is greater than or equal to -2 and less than 4.”

We represent B on a number line as:

B is an infinite set, and n(B) = ∞

B: Set Builder Notation

Example 2:

Solution to example 2:

C. Complement s of sets

The symbol U is used to represent the universal set under consideration.

Example:

Suppose we are only interested in the natural numbers from 1 to 20, and we want to consider subsets of this set. We say the set U = {x | 1 < x < 20, x e N } is the universal set in this

situation.

C. Complement s of sets

The complement of A, denoted A’, is the set of all elements of U which are not in A.

Example:

If the universal set U = {1, 2, 3, 4, 5, 6, 7, 8}, and the

set A = {1, 3, 5, 7, 8}, then the complement of A is

A’ = {2, 4, 6}.

C. Complement s of sets

Three obvious relationships are observed connecting A and A’.

1. A A’ = Ø as A’ and A have no common members.

2. A A’ = U as all elements of A and A’ combined make

up U.

3. n(A) + n(A’) = n(U)

C. Complement s of sets

Example 3:

C. Complement s of sets

Solution to example 3:

C. Complement s of sets

Example 4:

C. Complement s of sets

Solution to example 4:

C. Complement s of sets

Example 5:

C. Complement s of sets

Solution to example 5:

D. Venn Diagrams

Venn diagrams are often used to represent sets of objects, numbers, or things.

A Venn diagram consists of a universal set U represented by a rectangle.

Sets within the universal set are usually represented by circles.

Example:

D. Venn Diagrams

Example of a universe set, U = {2, 3, 5, 7, 8}, A = {2, 7, 8}, and

A’ = {3, 5}.

D. Venn Diagrams

SUBSETS

If B A then every element of B is also in A. The circle representing B is placed within the circle representing A.

INTERSECTION

A B consists of all elements common to both A and B.

It is the shaded region where the circles representing A and B

overlap.

D. Venn Diagrams

UNION

A B consists of all elements in A or B or both. It is the shaded region which includes both circles.

D. Venn Diagrams

DISJOINT OR MUTUALLY EXCLUSIVE SETS

Disjoint sets do not have common elements. They are represented by non-overlapping circles.

For example, if A = {2, 3, 8} and B = {4, 5, 9} then A B = Ø.

D. Venn Diagrams

Example 6:

Solution to example 6:

Example 7:

Solution to example 7:

E. Venn Diagram Region

The shading representations of Venn Diagrams.

Example 8:

Solution to example 8:

F. Numbers in Regions

The four regions of the Venn Diagram that contains two overlapping of sets A and B.

F. Numbers in Regions

Example 9:

F. Numbers in Regions

Solution to Example 9:

F. Numbers in Regions

Venn diagrams allow us to easily visualize identities such as

n(A B’) = n(A) – n(A B) and

n(A’ B) = n(B) – n(A B)

F. Numbers in Regions

Example 10:

Given n(U) = 30, n(A) = 14, n(B) = 17, and n(A B) = 6, find:

a. n(A B) b. n(A, but not B)

F. Numbers in Regions

Solution to example 10:

G. Problem solving with Venn Diagrams

Example 11:

A squash club has 27 members. 19 have black hair, 14 have

brown eyes, and 11 have both black hair and brown eyes.

a. Place this information on a Venn diagram.

b. Hence find the number of members with:

i. black hair or brown eyes

ii. black hair, but not brown eyes.

G. Problem solving with Venn Diagrams

Solution to example 11:

G. Problem solving with Venn Diagrams

Example 12:

A platform diving squad of 25 has 18 members who dive from 10 m and 17 who dive from 4 m. How many dive from both platforms?

G. Problem solving with Venn Diagrams

Solution to example 12:

G. Problem solving with Venn Diagrams

Example 13:

A city has three football teams in the national league: A, B, and C. In the last season, 20% of the city’s population saw team A play, 24% saw team B, and 28% saw team C. Of these, 4% saw both A and B, 5% saw both A and C, and 6% saw both B and C. 1% saw all three teams play.

Using a Venn diagram, find the percentage of the city’s population which:

a. saw only team A play

b. saw team A or team B play but not team C

c. did not see any of the teams play.

Solution to example 13:

Solution to example 13:

Solution to example 13: