115.

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 ≥ 7 }                                       (xi) {x : x ∈ N and 2x + 1 > 10}

∴ A ' = U - A = 
              
                 = {x : x  is an odd natural number}


(ii)  A = {x : x is an odd number}

A' = U - A =  - {x : x is an odd number}

= {x : x is an even natural number} = 


(iii) A = {x : x is a positive multiple of 3}

A' = U - A = {1, 2, 3, 4, 5, 6, .......} - {3, 6, 9, 12, .........}

= {1, 2, 4, 5, ...........} = , x is not a multiple of 3}


(iv)    A = {x : x is a prime number} = , x is priime number}

A' = U - A =  -  x is a prime number

= , x is not a prime number}


(v)    A = , x is divisible by 3 and 5}

=, x is divisible by 15.

A' = U - A  = , x is divisible by 15}

= , x is not divisible by 15}

=    , x is not divisible by 3 and 5}


(vi) A = { x : x is a perfect square}

A' = U - A =  - (x : x is a perfect square)

= , x is not a perfect square}


(vii)   A = {x : x is a perfect cube}

A' = U - A =  - {x : x is a perfect cube}

= , x is not a perfect cube}



(viii)  A = {x : x +5 = 8} = {x : x = 3} = {3}

∴   A' = U - A = {1, 2, 3, 4, 5, 6, .........} - {3} = {1, 2, 3, 4, 5, 6, .........}

∴   A' = 



(ix)    A = { x : 2x + 5 = 9} = {x : 2x = 4} = {x : x = 2} = {2}

∴   A' =  U - A = {1, 2, 3, 4, 5, 6, ..........} - {3} = {1, 2, 4, 5, 6, ............}





(x)     A = 

A' = U - A = 

= {1, 2, 3, 4, 5, 6}



(xi)       

∴    A' = U - A = {1, 2, 3, 4, 5, 6, 7, 8, .........} - {5, 6, 7, 8, ..........}
      = {1, 2, 3, 4}