Determine whether the below relations is reflexive, symmetric and transitive:
Relation R in the set A = {1, 2, 3, ..., 13, 14} defined as R = {(x, y) : 3x – y = 0}.
✓
Answer
The Relation R on the set A = {1, 2, 3, ...., 13, 14}, is defined as
R = {(x, y) : 3x - y = 0}
Then, R = {(1, 3), (2, 6), (3, 9), (4, 12)}
Reflexive
A relation is said to be reflexive if (x, x) $\in$ R, for every x in the domain.
Clearly, R is not reflexive as (1,1), (2,2) …… (14,14) $\notin$ R
Symmetric
A relation is said to be symmetric if (y, x) $\in$ R whenever (x, y) $\in$ R.
But here, R is not symmetric as (1,3) $\in$ R, but (3,1) $\notin$ R
Transitive
A relation is said to be transitive if (x, z) $\in$ R whenever (x, y) $\in$ R and (y, z) $\in$ R
But here, R is not transitive as (1,3), (3,9) $\in$ R, but (1,9) $\notin$ R
Therefore, R is neither reflexive, nor symmetric, nor transitive.
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