Chapter 2.2-2.3 Flashcards
Universal Set
U, is the set of all elements that are being considered
Complement
A’, the set of all elements that are in U and not in A
U’
∅
∅’
U
(A’)’
A
Well-defined
a set is well-defined if it is possible to determine whether any given item is or is not an element of the set
Set that is not well-defined
{x/x is a nice cat} because nice is ambiguous
Empty set or null set
is the set with no elements, denoted ∅
Finite
a set is finite if the number of elements in the set is a whole number
cardinal number or cardinality
the cardinality of a finite set A, denoted n(A), is the number of elements in A
Cardinality of the empty set
0
Set A is equal to set B if….
if and only if, A and B have exactly the same elements, denoted A=B
Set A is equivalent to set B if…
if and only if the cardinality of A, n(A), equals the cardinality of B, n(B), denoted A~B
Subset
Let A and B be two sets. A is a subset of B, denoted A ⊆ B if and only if every element of A is also in B
A= {1, 2, 3}
B= {1, 2, 3, 4, 5}
Proper Subset
Let A and B be sets, A is a proper subset of B, denoted A⊂B, if and only if A ⊆ B and A≠B
A= {1, 2, 3}
B= {1, 2, 3, 4, 5}
Is this a subset?
A= {5, 10, 44}
B= {5, 10, 44}
Yes
Is this a subset?
W⊆ ℕ
No
Is this a subset?
A’⊆A
No
Is this a proper subset?
A= {5, 10, 44}
B= {5, 10, 44}
No, because A=B
How to find number of subsets
If n(A)=K then A has 2^K subsets
How to find the number of proper subsets
If n(A)=K then A has 2^K - 1 proper subsets
Intersection
Let A and B be two sets, the intersection of A and B, denoted A∩ B, is the set of all elements that belong to both A and B
A∩ B= {x/xEA and xEB}
U= {1, 2, 3...10}
A= {1, 4, 5, 7}
B= {2, 3, 5, 10}
C= {3, 8, 10}
Find A∩ B{5}
U= {1, 2, 3...10}
A= {1, 4, 5, 7}
B= {2, 3, 5, 10}
C= {3, 8, 10}
Find B∩ C{3, 10}
U= {1, 2, 3...10}
A= {1, 4, 5, 7}
B= {2, 3, 5, 10}
C= {3, 8, 10}
Find A∩ C∅, A and C are disjoint
Union
let A and B be sets, the union of A and B, denoted AUB, is the set of all elements that belong to either A or B
AUB={x/xEA or xEB}
U= {1, 2, 3...10}
A= {1, 4, 5, 7}
B= {2, 3, 5, 10}
C= {3, 8, 10}
Find A U B{1, 2, 3, 4, 5, 7, 10}
U= {1, 2, 3...10}
A= {1, 4, 5, 7}
B= {2, 3, 5, 10}
C= {3, 8, 10}
Find A U C{ 1, 3, 4, 5, 7, 8, 10}
U= {1, 2, 3...10}
A= {1, 4, 5, 7}
B= {2, 3, 5, 10}
C= {3, 8, 10}
Find B U C{2, 3, 5, 8, 10}
A U ∅
A
U= {1, 2, 3, 4...10}
A= {2, 4, 6, 8}
B= {3, 6, 9}
C= {1, 4, 8}
Find A U (B ∩ C){2, 4, 6, 8}
U= {1, 2, 3, 4...10}
A= {2, 4, 6, 8}
B= {3, 6, 9}
C= {1, 4, 8}
Find B ∩ C'{3, 6, 9}
U= {1, 2, 3, 4...10}
A= {2, 4, 6, 8}
B= {3, 6, 9}
C= {1, 4, 8}
Find [A ∩ (B U C)]'B U C= {1, 3, 4, 6, 8, 9}
A ∩ (B U C) = {4, 6, 8}
[A ∩ (B U C)]’ = {1, 2, 3, 5, 7, 9, 10}
DeMorgan’s Laws
For all sets A and B:
(A U B)’ = A’ ∩ B’
(A ∩ B)’ = A’ U B’
Commutative Property
A U B = B U A
A ∩ B = B ∩ A
Associative Property
A ∩ (B ∩ C) = (A ∩ B) ∩ C A U (B U C) = (A U B) U C
Distributive Property
A ∩ (B U C) = (A ∩ B) U (A ∩ C) A U (B ∩ C) = (A U B) ∩ (A U C)
