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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: