Question

State the number of subsets of {p, q, r, s}. Explain the answer please as I don't understand how to do this.

Answer 1

Answer is $2^4=16$.

Explanation :

We calculate the number of subsets of the given set according to the number of elements present in the subset .

Case 1 :

The number of subsets of cardinality 0 ( cardinality means the number of elements in the given set ) = 1$=^4C_0$

Because the only subset of cardinality 0 is $\phi$ .

Case 2 : the number of subsets of cardinality 1 = $^4C_1=4$

(You can also observe $\{p\} , \{q\},\{r\},\{s\}$ are the only subsets of cardinality 1 of the given set ).

Case 3 : the number of subsets of cardinality 2 $=^4C_2$ $=6$

( You can also observe that $\{p,q\},\{p,r\},\{p,s\},\{q,r\},\{q,s\},$

$\{r,s\}$ are the only subsets of cardinality 2 of the given set.)

Case 4 : the number of subsets of cardinality 3 $=^4C_3=4$

(You can also observe that

$\{p,q,r\},\{p,q,s\},\{p,r,s\},\{q,r,s\}$

are the only subsets of cardinality 4 of the given set ).

Case 4 : the number of subsets of cardinality 4 $=^4C_4=1$

(Observe that only subset of cardinality 4 of the given set is the set itself ).

Hence the total number of subsets of the given set $=^4C_0+^4C_1+^4C_2+^4C_3+^4C_4$

$=(1+1)^4$

(Using binomial theorem )

$=2^4=16$

(You can observe that

$1+4+6+4+1=16$ )