Test by Bikram | Mock GATE | Test 3 | Question: 27

Question

Let $A$ be the set of non-zero integers.

Let $R$ be the relation on $K × K$ defined as $\left ( a,b \right )$$R$ $\left ( c,d \right )$  if  $ad = bc$.

The relation $K$ is a/an:

• A.    Equivalence Relation
• B.    Poset
• C.    Antisymmetric
• D.    Reflexive and symmetric, but not transitive

Answer 1

1. If we have (a, b) R (a, b) since ab = ab. The relation is reflexive

2. If we have (a, b) R (c, d) then we have ad = bc. Accordingly, bc = ad yields (c, d) R (a, b). The relation is symmetric.

3. If we have (a, b) R (c, d) and (c, d) R (e, f), we get ad = bc and cf = de.
This implies,
(ad)(cf) = (bc)(de) => af = be

  By canceling cd from both sides we got the last result .