We consider the relation 𝑅={(𝑎,𝑎),(𝑎,𝑏),(𝑎,𝑐),(𝑎,𝑑),(𝑎,𝑒),(𝑏,𝑐),(𝑏,𝑒),(𝑑,𝑎),(𝑒,𝑐)}.Is 𝑅 transitive? Justify briefly
Question
We consider the relation R = {(a, a), (a, b), (a, c), (a, d), (a, e), (b, c), (b, e), (d, a), (e, c)}. Is R transitive? Justify briefly.
Solution
A relation R on a set A is said to be transitive if for every a, b, c in A, whenever (a, b) and (b, c) are in R, then (a, c) is also in R.
Let's check the given relation R = {(a,a),(a,b),(a,c),(a,d),(a,e),(b,c),(b,e),(d,a),(e,c)} for transitivity.
We can see that (a, b) and (b, c) are in R, and (a, c) is also in R. Similarly, (a, d) and (d, a) are in R, and (a, a) is also in R.
However, we have (d, a) and (a, b) in R, but (d, b) is not in R. Similarly, we have (b, e) and (e, c) in R, but (b, c) is not in R.
Therefore, the relation R is not transitive.
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