Question

Give a relation R={(1,2), (2,3)} on the set of natural numbers, add a minimum number of ordered pairs.

• A. enlarged relation is symmetric
• B. enlarged relation is transitive
• C. enlarged relation is reflexive.
• D. enlarged relation is equivalence relation

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A enlarged relation is reflexive.
B enlarged relation is symmetric
C enlarged relation is transitive
D enlarged relation is equivalence relation

$R=\left\{\right(1,2),(2,3\left)\right\}$
A)To make the relation symmetric, we have to add (2,1) and (3,2) in R.
Enlarged symmetric relation $R^{\prime }=\left\{\right(1,2),(2,3),(2,1),(3,2\left)\right\}$
B) To make enlarged relation transitive , we have to add (1,3), (1,1),(2,2) in R'.
Also, we need to add (3,1) to make it symmetric
So, the new enlarged relation is $R^{\prime \prime }=\left\{\right(1,2),(2,3),(2,1),(3,2),(1,1),(2,2),(3,1\left)\right\}$
C) For enlarged relation to be symmetric, we need to add (1,1) in R''
$R^{\prime \prime }=\left\{\right(1,2),(2,3),(2,1),(3,2),(1,1),(2,2),(3,1),(1,1\left)\right\}$
D) For enlarged relation to be equivalence,