Taking the set of natural numbers as the universal set, write down the complements of the following sets :
(i) {x : x is an even natural number}
(ii) {x : x is an odd natural number}
(iii) {x : x is a positive multiple of 3}
(iv) {x : x is a prime number}
(v) {x : x is a natural number divisible by 3 and 5}
(vi) {x : x is a perfect square}
(vii) {x : x is a perfect cube}
(viii) {x : x + 5 = 8}
(ix) {x : 2x + 5 = 9}
(x) `{x : x ge 7}`
(xi) `{x : x in N and 2x + 1 gt 10}`.

Explanation

U = set of natural numbers (N)
`={1,2,3,4,5,6,7,8,9,10,....}`
(i) Let {x : x is an even natural numbers} = A
`rArrA={2,4,6,8,10,....}`
`:. A'=U-A`
`={1,3,5,7,9,....}`
= set of odd natural numbers
(ii) Let {x : x is an odd natural number} = A
`rArr A={1,3,5,7,....}`
`:. A'=U-A`
`= {2,4,6,8,10,...}`
= set of even natural numbers
(iii) Let {x : x is a positive multiple of 3} = B
`B = {3,6,9,12,15}`
`:. B'={1,2,4,5,7,8,10,11,....}`
= {x : x `in N` , x is not a multiple of 3}
(iv) Let {x : x is a prime number} = A
`:. A' = U - A`
= {x : x is a composite natural number or x = 1}
(v) The complement of set {x : x, is a natural number divisible by 3 and 5} is the set of those natural numbers
which are not divisible by 3 and 5 both
`:.` Complement set = {x : x `in N` , x is not divisible by 3 and 5 both}
(vi) The complement of set {x : x `in N` , is a perfect square} is {x : x `in N` is not a perfect square}
(vii) The complement of set {x : x is a perfect cube is {x : x is a not a perfect cube}
(viii) Given set `{x : x + 5 = 8}`
`because x + 5 = 8 rArr x =3`
`:. (x : 5} = {3}`
then complement of `{x : x + 5 = 8} = U - {3}`
`= N - {3}`
`= {x : x in N , x ne 3}`
(ix) Given set `{x : 2x + 5 = 9}`
Now, `2x + 5 = 9 rArr 2x = 4 rArr x=2`
`:. {x : 2x + 5 = 9} = {2}`
then, complement of `{x : 2x 5 =9} = U - {2}`
`= N - {2} = {x :x in N , x ne 2}`
(x) Given set `{x : x ge 7}`
`because {x:x ge}={7,8,9,10,11,12,...}`
`:.` Complement of `{x : x ge 7}=` set of natural numbers in which no element is 7 or greater than 7.
= set of natural number smaller than 7
`= {x : x in N,x lt7}`
Therefore, complement of `(x:x ge7)={x:x in N,x lt7)or{1,2,3,4,5,6}`
(xi) Given set `{x:x in N and 2x+1gt10}`
`because 2x+1 gt10rArr2x gt9rArrx gt (9)/(2)`
Then complement of set {`x : x sub N and 2x + 1 gt 10`}
`= U-{x:x in N,2x + 1 gt10}`
`=N-{x:x in N,x gt(9)/(2)}`
`={1,2,3,4}`.