Question

Let the universal set be U - {x:x element of N and x < or equal to 18), and A={factors of 12) and B={factors of 18}. Find the following(i) A'<br />(ii) B'(iii) A'UB'(iv) A'nB'.Also, verify that (a) (AUB')= A'n B' and (b) (A'nB)'=A'UB'​

Answer 1

Step-by-step explanation:

U={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}

A={1, 2, 3, 4, 6 , 12}

(i)A'=U -A

={1,2,3,4,5....18 }-{1, 2, 3, 4, 6 ,12},

={5,7,8,9,10,11,13,14,15,16,17,18}

B={1, 2, 3, 6, 9, 18}

(ii)B'=U -B

={1,2,3,4,5,6,7....18}-{1, 2, 3, 6, 9, 18}

={4,5,7,8,10,11,12,13,14,15,16,17}

A ∪ B={1, 2, 3, 4, 6 , 12}∪{1, 2, 3, 6, 9, 18}

={1,2,3,4,6,12,18}

(iii)A' ∪ B'= U -(A ∪ B)

={1,2,3,4,5,....18} -{1,2,3,4,6,12,18}

={5,7,8,9,10,11,13,14,15,16,17}

(iv)A'∩ B'={1, 2, 3, 4, 6 , 12}∩{1, 2, 3, 6, 9, 18}

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

we know,

A∪B={1,2,3,4,6,12,18}

(A∪B)'= U -(A∪B)

={1,2,3,4,5...18}-{1,2,3,4,6,12,18}

={5,7,8,9,10,11,13,14,15,16,17}

(a) in this question its (A∪B)'=A'∪B'

so,

(A∪B)'=A'∪B' proved

we know,

A'∩ B'{1,2,3,4,6}

(A∩ B)'={1, 2, 3, 4, 6 , 12} ∩ {1, 2, 3, 6, 9, 18}

={1,2,3,6}

(b) in this question its (A∩B}'=A'∩B'

so,

(A∩B}'=A'∩B' proved.

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Answer 2

Step-by-step explanation:

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