Write down :

(i) The set C of letters of the word 'PAPAYA'.

(ii) All subsets of C.

(iii) All proper subsets of C.

(i) The set C of letters of the word 'PAPAYA'.

In roster form, every repeated element in a set is taken only once.

The distinct letters in the word 'PAPAYA' are P, A, and Y.

C = {P, A, Y}

(ii) All subsets of C.

The empty set (Φ) is a subset of every set.

Every set is a subset of itself.

Since set C has 3 elements, it will have 2³ = 8 subsets.

Subsets are: Φ, {P}, {A}, {Y}, {P, A}, {A, Y}, {P, Y}, {P, A, Y}

(iii) All proper subsets of C.

A proper subset includes all subsets of the set except for the set itself.

There are 2ⁿ - 1 proper subsets, which means 8 - 1 = 7 for this set.

Proper subsets are: Φ, {P}, {A}, {Y}, {P, A}, {A, Y}, {P, Y}