Question

Set operations including cardinality of power sets and listing the elements of a power set and finding the number of subsets of a given size. let u = {-1, 0, 1, 2, 3, 4, 6} , a = {-1, 1, 2, 3, 4} and b = {0, 2, 4, 6}. find

a.a u b,
b.a ∩ b,
c.b – a,
d.the power set of b and its cardinality

Answer 1

a. a
`\cup` b
a
`\cup` b means the resulting set when a and b are merged, less duplicates.
a
`\cup` b={-1, 1, 2, 3, 4}
`\cup` {0, 2, 4, 6}
={-1,0,1,2,3,4,6}

b. a
`\cap` b
a
`\cap` b means the set of members which are common to both a and b.
a
`\cap` b={-1, 1, 2, 3, 4}
`\cap` {0, 2, 4, 6}
={2,4}

c. b-a
b-a is the set of members present in b but not in a.
{0, 2, 4, 6}-{-1, 1, 2, 3, 4}
={0,6}

d. Power set of b, its cardinality
Power set is the set of all possible combinations of elements.
There are 2^n members in the power set of x where n is the number of elements in the set x.
For example, power set of S={1,2} is {
`\phi`,{1},{2},{1,2}}
The cardinality of S above is the number of elements in the set, therefore 2^2=4.

For b={0,2,4,6}
The power set is
{
`\phi`,{0},{2},{4},{6},{0,2},{0,4},{0,6},{2,4},{2,6},{4,6},{0,2,4},{0,2,6},{0,4,6},{2,4,6},{0,2,4,6}}
for a total of 2^4=16 members in the power set.
Therefore the cardinality of the power set is 16.