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Question

If n(A)=n,n>0 then which among the following statements are correct

A
Number of relations on A that are not reflexive =(2n2n)(2n1)
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B
Number of relations on A that are not reflexive =(2n2+n)(2n1)
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C
Number of relations on A that are not symmetric
=2n22n(n+1)2
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D
Number of relations on A that are not symmetric
=2n22n(n1)2
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Solution

The correct option is C Number of relations on A that are not symmetric
=2n22n(n+1)2
If n(A)=n
Let A={a1,a2,a3,.,an}
A×A=⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪(a1,a1),(a1,a2),,(a1,an)(a2,a1),(a2,a2),,(a2,an)....(an,a1),(an,a2),,(an,an)⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
n(A×A)=n2
Let us Consider elements in A×A as
(1)Diagonal elements D1={(a1,a1),(a2,a2)...(an,an)}
(2)Non diagonal elements D2={(a1,a2),(a2,a1),(a1,a3),(a3,a1)...(an1,an),(an,an1)}
n(D1)=n,n(D2)=n2n
Now for relation to be reflexive,
(i) there is only one choice for diagonal elements i.e It has to be included to make relation reflexive
(ii) for each of n2n non diagonal elements, there are two choices either it can be included or excluded to make relation reflexive.
from principle of counting, number of reflexive relations
=1×1×...×1(n times)×2×2×...×2(n2n times)
=2n2n
So, number of reflexive relations=2n2n
number of relations which are not reflexive =2n22n2n=2n2(112n)=2n2n(2n1)

Now for relation to be symmetric,
(i) there are two choices for each of n diagonal elements
(ii) For n2n non diagonal elements we can form pairs of (a1,a2),(a2,a1);(a1,a3)(a3,a1);(an1,an)(an,an1);
for each pair there are two choices, either it can be included or excluded to make relation symmetric.
From principle of counting number of symmetric relations
=2×2×...2(n times)×2×2×...2(n2n2 times)=2n2+n2
So, number of symmetric relations =2n2+n2
number of relations which are not symmetric =2n22n2+n2

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