# 2.2 Venn Diagrams and Subsets

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```					2.2 Venn Diagrams and
Subsets
• Venn diagrams (developed by the John
Venn) are drawings represented by a
rectangle and oval regions.
– Rectangle: the universe
– Oval regions: subsets of the universe
• Universe (represented as U) includes all
things under discussion at a given time.
– Ex: When we discuss about the numbers of
people who can speak Spanish at Loudoun
campus. The universe can be everybody at
the school include students, faculties and
staffs. The subset is the set of people who
can speak Spanish
• The complement of a set A, symbol A’ (read A
prime)
A’ = {x | x Є U and x Є A}

• Ex1): Let U = {1,2,3,4,5}             1,2,4, 3,5
A
A = {1,2,4}
A’
A’ = ?
A’ = {3,5}
Ex2) What is the complement of the empty set?
{x | x Є U and x Є }
• Set A is a subset of set B (A     B) if every element
of A is also an element of B
Ex: {1,2,3}       {1,2,3,4}
{}        U
{a, b}       {a, b}

• Set A is equal to set B (A = B) if A   B and B     A
(**every set is a subset of itself )

• Set A is a proper subset of set B (A     B) if A   B
and A ≠ B
Ex: {1,2}    {1,2,3,4}
Ex: Find all possible subsets of each set
1) { }
Subset is:
2) {a}
Subsets are:
3) {a, b}
Subsets are:
4) {a,b, c}
Subsets are:
If a set has 0 element - we have 1 subset
1          -        2
2         -         4
3         -         8
4         -         ?
n         -         ?

Therefore:
if a set has n element, there are 2n subsets
or 2n - 1 proper subsets

```
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