UGC NET CSE | December 2019 | Part 2 | Question: 8

Question

How many reflexive relations are there on a set with $4$ elements?

• A. $2^4$
• B. $2^{12}$
• C. $4^2$
• D. $2$

Answer 1

Understanding reflexive relation using matrix.

Let us have n elements in a set.

In a n*n matrix we can have 2 possibilities that the relation is present (represent as 1) or not present (represent as 0).

A matrix is reflexive if all the diagonal elements are 1.

$\begin{bmatrix} 1 & & \\ & 1 & \\ & & 1 \\ & & & 1 \end{bmatrix}$

Now we are left with n$^{2}$-n space to be filled with either 0 or 1,

So total number of Reflexive relations will be 2$^{n^{{2}}-n}$.

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Here n=4

So 2$^{4^{{2}}-4}$ = 2$^{16-4}$ = 2$^{12}$

Answer 2

A has 4 elements.So, $A*A$ has 16 elements.

let us take a example. Let A ={a,b,c,d}

$A*A=\{aa,bb,cc,dd,ab,ac,ad,ba,bc,bd,ca,cb,cd,da,db,dc\}$

Here iam representing ordered pair (a,b) with ‘ab’

Ordered pairs aa,bb,cc,dd has no choice they have to be present in every reflexive relation.

But remaining 12 ordered pairs have 2 choices:Present or not present in a reflexive relation.

With each of these12 ordered pairs having 2 choices, we have $2^{12}$ relations each of which is reflexive.