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Union Symbol ∪
If A and B are sets, their union is equal to all
elements in both A & B
A = {1,2,3,4}
B = {2,4,5,6,7,8}
A ∪ B = {1,2,3,4,5,6,7,8}
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Venn Diagram
Square with U
The U indicates the universal set. The
universal set is all objects in a particular
discussion.
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The intersection ∩ is the set of all elements
that belong to both A & B.
A ={1,2,3,4}
B = {2,4,5,6,7,8}
A & B have the numbers 2 and 4 in common
A∩B = {2,4}
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Disjoint sets are sets with no common
elements.
A = {1,2,3}
B = {4,5,6}
A ∩ B = {}
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Compliment
If A and B are 2 sets, the compliment of B,
with respect to A, is the set of all elements
that belong to A but not to B.
Compliment of B = A – B denoted B
Compliment of A = B – A denoted A
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The universal set includes A
The compliment of A is denoted A and is
indicated by the red shaded area.
The complement of A with regard to the
universal set.
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Symmetric Difference ⊕
S = {a,b,c,d}
T= {a,c,e,f,g}
S⊕T = {b,d,e,f,g}
What is in S that is not in T
What is in T that is not in S
Formula S⊕T = (S-T) ∪ (T-S)
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Cardinality of a finite set is the number of
distinct elements in a set. The cardinality of
a finite Set A is denoted |A|.
A = {a,b,c}
|A| = 3
There are 3 distinct elements in set A
A finite set has a limited size.
An infinite set has no size limitation.
Be familiar with the properties on pages 8 & 9
 Cummutative
 Associative
 Distributive
 Idempotent
 Properties of the compliment
 Properties of a universal set
 Properties of the empty set
The Addition Principle
1. Disjoint
|A ∪ B| = |A| + |B|
2. Find A ∪ B
Cardinality of A = 10
Cardinality of B = 7
A ∩B = 4
A ∪ B counts the intersection twice, therefore,
subtract A ∩B .
| A ∪ B |=|A|+|B|-| A ∩B |
| A ∪ B |=10 + 7 – 4
| A ∪ B |= 13
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The formula for three sets
|A∪B∪C|=|A|+|B|+|C|-|A∩B|-|B∩C|-|A∩C|+|A∩B∩C|
People were asked what mode of transportation they used: Bus,
Train, Automobile. Can select any or all.
|B| = 30, |T| = 35, |A| = 100
|B ∩ T| = 15, |B ∩ A| = 15, |T ∩ A| = 20, |B ∩T ∩A| = 5
|B∪T∪A|=|B|+|T|+|A|-|B∩T|-|B∩A|-|T∩A|+|B∩T∩A|
|B∪T∪A|=30+35+100-15 -15 - 20 +5
|B∪T∪A|=120