Question

The number of subsets of $\left\{ {1,2,...,9} \right\}$ containing at least one odd number is
a.324
b.396
c.496
d.512

Answer 1

Hint: The number of elements in the given set is 9 and number of subsets of a set with n elements is given as ${2^n}$ .Hence the number of subsets of the given set is given as 512 and the number of subsets with only even numbers is 16 . All the other subsets of $\left\{ {1,2,...,9} \right\}$ other than the subsets of $\left\{ {2,4,6,8} \right\}$contain at least one odd number .Hence the difference gives the required number.

Complete step-by-step answer:
We are given a set $\left\{ {1,2,...,9} \right\}$
We can see that the number of elements in the set is 9
We know that the number of subsets of a set with n elements is given as ${2^n}$
Hence the number of subsets of the given set is given as
$ \Rightarrow {2^9} = 512$
Now the subset of even numbers from the given set is $\left\{ {2,4,6,8} \right\}$
Now let's find the number of subsets formed from the set $\left\{ {2,4,6,8} \right\}$
The number of elements in this set is 4
Hence the number of subsets of the given set is given as
$ \Rightarrow {2^4} = 16$
All the other subsets of $\left\{ {1,2,...,9} \right\}$ other than the subsets of $\left\{ {2,4,6,8} \right\}$ contain at least one odd number
Therefore the number of subsets of $\left\{ {1,2,...,9} \right\}$ with at least one odd number is given by
$ \Rightarrow 512 - 16 = 496$
Therefore the correct option is c

Note: In set theory, a subset is a set which has some (or all) of the elements of another set, called superset, but does not have any elements that the superset does not have.
A subset which does not have all the elements of its superset is called a proper subset.