Question

Consider the sets A={n e P:n is odd} B = {n e P:n is prime) C = {4n +3:n € P} D = {x e R : x2 - 8x + 15 = 0} Which of these sets are subsets of which? Consider all 16 possibilities.

Answer 1

`A\subset A, B\subset B,C\subset C,D\subset D`,
`A\subset B`,
`D\subset A` and
`D\subset B`.

Explanation:

The given sets are

A={n ∈ P:n is odd}

B = {n ∈ P:n is prime)

C = {4n +3:n ∈ P}

D = {x ∈ R : x² - 8x + 15 = 0}

P is the set of prime numbers and R is the set of real numbers.

`x^2 - 8x + 15 = 0`

`x^2 - 5x-3x + 15 = 0`

`x(x-5)-3(x-5) = 0`

`(x-5)(x-3) = 0`

So, the elements of all sets are

A = {3, 5, 7, 11, 13, 17, 19, 23, ...}

B = {2, 3, 5 , 7, 11, 13, 17, 19, 23, ...}

C = {11, 15, 23, ...}

D = {3,5}

Each sets is a subset of it self. So,

`A\subset A, B\subset B,C\subset C,D\subset D`

All the elements of A lie in set B, so A is a subset of B.

`A\subset B`

Since
`15\\otin A` and
`15\\otin B`, So, C is not the subset of A and B.

D has two elements, 3 and 5, Since
`3,5\in A` and
`3,5\in B`, therefore

`D\subset A` and
`D\subset B`