Transitive relation

Step 1: ((1,2)) and ((2,4)) imply ((1,4)). Step 2: Then ((1,4)) and ((4,8)) imply ((1,8)). Step 3: Apply transitivity repeatedly along the chain.

Step 1: Transitivity is not checked by counting pairs; it is checked by finding linked pairs. Step 2: For every ((a,b)) and ((b,c)), ((a,c)) must be present. Step 3: Making a small table reduces mistakes in exams.

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.