2.
2 SETS
2.2 SETS
2. Subset - a set (say, A) is a subset of another set (say, B),
written as A ⊆ B , if and only if, every element of A
is also an element of B.
(Alternative way: A is contained in B or B contains A)
# of subsets of B is 2n where n is the cardinal number of set B
Proper subset,
Def’n: A B = { x | ((x A) → (x B) (A B) }
Subset, ⊆
Def’n: A ⊆ B = { x | ((x A) → (x B) (A may be = B) }
EX 1. List all the possible subsets of set D = {1, 2, 3}.
How many subsets are there in all?
There should be 8 SUBSETS in all, namely:
Ø or { } {1} {2} {3}
{ 1, 2 } { 1, 3 } { 2, 3 } { 1, 2, 3 }
EX 2. Refer to Ex 1. Which subsets of D are PROPER?
EX 3. IF set F contains 5 elements, how many subsets should it have?
EX 4. Refer to EX 3. How many of the subsets are proper?
Summary of important sets:
ACTIVITY 1: Find the following:
1. The set of all odd integers
2. { x Z | − 2 x 5}
3. { x Z + | − 2 x 5}
4. { x R | − 2 x 5}
5. { n Z | n = 2 p, for some integer p }
6. { x Z + | x is odd and x 20 }
Common Set Notations
Symbol Read as Given Example
element of Let A = {0, 2, 4, 6, 8}. 2A
not an element of 1 A
subset of {2, 4} A
not subset of {2, 5} A
∩ intersection Let B = {1, 2, 3, 4} A ∩ B = { 2, 4 }
∪ union A ∪ B = {0, 1, 2, 3, 4, 6, 8}
ˊ complement If universal set, U = {0,1, 2, …, 9}, then, Aˊ = {1, 3, 5, 7, 9}
empty Let C = {5, 7} . B ∩ C = { } or
\ minus or symmetric difference A \ B = {0, 6, 8}; B \ A = {1, 3}
Definitions
{ x U | x A and x B }
Set operations and Venn Diagrams
ASSIGNMENT #1: Illustrate #s 1 – 8 using Venn diagrams. (SHADE lightly.)
ACTIVITY 2. Find the following:
ASSIGNMENT #2
ASSIGNMENT #3
4. Cartesian product of sets
For 2 sets, A and B, the Cartesian product of A and B is
AxB = the set of ALL ordered pairs (x, y) such that
x comes from set A and y comes from set B
For 3 sets, say A, B, and C:
A x B x C = the set of ALL ordered triples (x, y, z) such that
x comes from set A, y comes from set B,
and z comes from set C.
ACTIVITY 3
ACTIVITY 1 (ANSWERS)
1. The set of all odd integers = { … , -3, -1, 1, 3, … }
2. { x Z | − 2 x 5} = { -1, 0, 1, 2, 3, 4}
3. { x Z + | − 2 x 5}= {1, 2, 3, 4}
( ) OR, in interval notation: (-2, 5)
4. { x R | − 2 x 5} = -2 5
5. { n Z | n = 2 p, for some integer p } = { … , -4, -2, 0, 2, 4, … }
6. { x Z + | x is odd and x 20 } = {1, 3, 5, … , 17, 19 }
ACTIVITY 2. (ANSWERS)
18
31
16
29
13
32 12 21
6
6 4
19
32
21
19
ACTIVITY 3 (SOME ANSWERS)
{(a, 0), (a, 1), (a, 2), (b, 0), (b, 1), (b, 2)}
{(0, a), (1, a), (2, a), (0, b), (1, b), (2, b)}
{(0, a, x), (0, b, x), (1, a, x), (1, b, x), (2, a, x), (2, b, x)}
= |B| x |A| x |C| = 3 x 2 x 1 = 6
NO! (Why?)
C x C x C or C3 = {(x, x, x)} ; |C x C x C| = 1
