Math Investigations
Name: ________________________________
Date: ______________ Block: _________
2.2: Venn Diagrams and Subsets (Part 1)
Objectives: 1. To use Venn diagrams to depict set relationships
2. To determine whether one set is a subset of another
3. To understand the difference between a subset and a proper subset
4. To determine the number of subsets of a given set
VENN DIAGRAMS & THE COMPLEMENT OF A SET
Key Terms:
~ Universal Set: _____________________________________________________________________________________
_________________________________________________________________________________________________________
~ The Complement of a Set
For any set A within a universal set U, the complement of A, written as A’, is the set of
elements of U that are not elements of A.
~ Venn Diagrams:
U=_____________________________________________________
A=_____________________________________________________
A’=____________________________________________________
Example 1: Finding Complements
Let U (Universal Set) = {a, b, c, d, e, f, g, h} M = {a, b, e, f} and
(a) Find the set M’
Quick Check 1:
Let U = {5, 10, 15, 20, 25, 30, 35, 40}
(a) Find the set P’
N = {b, d, e, g, h}
(b) Find the set N’
K = {10, 20, 30, 40} and
(b) Find the set K’
P = {15, 5, 35}
 Consider the compliment of the Universal Set, U’. The set U’ is found by selecting all the
elements of U that do not belong to U.
*** There are no such elements, so there can be no elements in the set U’.
Therefore U’ = ____________
SUBETS OF A SET
 Suppose that we are given universal set U = {1, 2, 3, 4, 5}, while A = {1, 2, 3}.
~ Every element of set A is also an element of the universal set U.
Therefore, set A is a SUBSET of U  _______________
 If “A is not a subset of U”  ________________
Example 2: Determining If One Set Is a Subset of Another
Write ⊆ or ⊈ in each blank to make a true statement.
(a) {3, 4, 5, 6} _______ {3, 4, 5, 6, 8}
(b) {a, d, m} _______ {a, d, m, v, z}
(c) {5, 6, 7, 8} _______ {6, 5, 8, 7}
Quick Check 2:
Write ⊆ or ⊈ in each blank to make a true statement.
(a) {x, y, z} _______ {a, b, y, z}
(c) {2, 32} _______ {2, 4, 8, 16, 32, 64}
Complete textbook pg. 58 #5 – 12 below
(b) {1, 2, 6} _______ {2, 4, 6, 8}
Math Investigations
2.2: Venn Diagrams and Subsets (Part 2)
Name: ________________________________
Date: ______________ Block: _________
Objectives: 1. To use Venn diagrams to depict set relationships
2. To determine whether one set is a subset of another
3. To understand the difference between a subset and a proper subset
4. To determine the number of subsets of a given set
PROPER SUBSETS
Suppose that we are given the following sets.
B = {5, 6, 7, 8} and A = {6, 7}
A is a subset of B, but A is not all of B. There is at least one element in B that is not in A.
In this case A would be called a PROPER SUBSET of B  _______________________
 In summary, Set A is a proper subset of set B if 𝐴 ⊆ 𝐵 and 𝐴 ≠ 𝐵; written 𝐴 ⊂ 𝐵.
Example 3: Determining Subsets and Proper Subsets
Decide whether ⊂, ⊆, or both could be placed in each blank to make a true statement.
(a) {5, 6, 7} ________ {5, 6, 7, 8}
(b) {a, b, c} ________ {a, b, c}
Quick Check 3: Determining Subsets and Proper Subsets
Decide whether ⊂, ⊆, or both could be placed in each blank to make a true statement.
(a) {x, y, z} ________ {z, y, x}
(b) {3, 6} ________ {-9, -6, -3, 3, 6, 9}
(c) {2, 3, 4} _______ {5, 6, 7}
Complete pg. 58 #13 – 20 below
*** Every set (except ∅) has at least two subsets, _______________________________________________
Example 4: Listing All Subsets of a Set
Find all possible subsets of each set.
(a) {7, 8}
(b) {a, b, c}
Quick Check 4: Listing All Subsets of a Set
Find all possible subsets of each set.
(a) {blue, white}
(b) {5, 10, 15}
COUNTING SUBSETS
Formulas:
 Finding Number of Subsets
_______________________
 Finding Number of Proper Subsets
_____________________________
Example 5: Finding the Number of Subsets and Proper Subsets
Find the number of Subsets and Proper Subsets for each set.
(a) {3, 4, 5, 6, 7}
(b) {x | x is an even integer between -7 and 9}
Quick Check 5: Finding the Number of Subsets and Proper Subsets
Find the number of Subsets and Proper Subsets for each set.
(a) {x | x is a positive whole number between 2 and 12, inclusive}
(b) {1, 2, 3, 4, 5, 9, 12, 14}
Complete pg. 59 #37 – 40 below