Transitive relation

The governing definition is transitivity: whenever (a,b) belongs to R and (b,c) belongs to R, the pair (a,c) must also belong to R. In this relation, (1,2) and (2,3) are present. The second component of the first pair equals the first component of the second pair, so they form a valid chain. Transitivity therefore requires (1,3), and that pair is indeed listed in R. Hence option A correctly demonstrates the property. The presence or absence of isolated pairs such as (1,1), (3,1), or (2,2) does not by itself establish this particular transitive implication.

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.

The governing concept is transitivity. A relation is transitive if (a,b) and (b,c) in R require (a,c) to be in R as well. Here the first two given pairs form a chain: (1,2) and (2,4). Matching the second component of the first pair with the first component of the second gives a required conclusion of (1,4). That pair is present in R, so the displayed chain passes the transitive check. Therefore option B is correct. Pair (4,1) would concern a reverse direction not supplied by the chain; (2,1) is not the required outer pair; and (4,4) does not follow from the two given pairs. The calculation is about connecting the first element to the final element.