Transitive relation

Correct answer: A. (1,2) and (2,3) imply (1,3)

The governing definition of transitivity is: if (a,b)∈R and (b,c)∈R, then (a,c) must also belong to R. In the given relation, (1,2) and (2,3) are both present. They form a chain because the second coordinate of the first pair, 2, equals the first coordinate of the second pair. The transitive conclusion is therefore (1,3), and this pair is indeed included in R. Hence option A correctly demonstrates the property. The presence of (1,1), (3,1), or (2,2) alone does not show the required implication; such facts may concern other relation properties, but they do not establish transitivity.

Correct answer: A. (1,2)

For transitivity, use the pattern (a,b) and (b,c), then require (a,c). Here the first given pair is (1,2), so a=1 and b=2. The second pair is (2,2), so b=2 and c=2. Matching the common middle component is the key step. Therefore the required conclusion is (a,c)=(1,2), which is option A. That pair is already present in R, so this particular chain satisfies the transitive requirement. Option B reverses the ordered pair, while options C and D do not use the first and last components of the chain in the required way. This checks one implication; a complete proof would inspect every applicable chain.