Transitive relation

Step 1: To show non-transitivity, find a chain where the required pair is missing. Step 2: In the first option, ((1,2)) and ((2,3)) are present, but ((1,3)) is not. Step 3: Recognising a counterexample gives quick marks.

Step 1: The transitivity condition becomes relevant only when both ((a,b)) and ((b,c)) are present. Step 2: If no such chain exists, the condition is not violated. Step 3: Do not quickly call a relation non-transitive when no chain exists.

Step 1: From ((2,3)) and ((3,5)), ((2,5)) is required. Step 2: ((2,5)) is given, and ((5,5)) creates no missing pair. Step 3: Check the main chain first, then inspect the remaining pairs.

Step 1: From ((2,5)) and ((5,6)), ((2,6)) is required. Step 2: This pair is not present in the given relation, so it must be added. Step 3: Also consider chains formed by already implied paths.

Step 1: From ((1,2)) and ((2,2)), ((1,2)) is required. Step 2: ((1,2)) is present in the relation, so the condition is satisfied. Step 3: A self-pair can require the same pair again.

Step 1: From ((2,1)) and ((1,1)), ((2,1)) is required. Step 2: ((2,1)) is already present, so transitivity is not broken. Step 3: Form the chain and match the outer pair in the original list.

Step 1: From ((1,2)) and ((2,3)), ((1,3)) is immediately required. Step 2: Other chains may also require pairs, but among the given options ((1,3)) is the correct required pair. Step 3: Derive required pairs chain by chain.

Step 1: Here all possible ordered pairs of (A={1,2,3}) are present. Step 2: Since every pair is present, every chain has its required pair. Step 3: A full Cartesian product relation is always transitive.

Step 1: From ((2,4)) and ((4,1)), ((2,1)) is required. Step 2: ((2,1)) is not given, so the relation is not transitive. Step 3: Check all chains, not just the first visible one.

Step 1: In ((x,y)) and ((y,z)), the middle element is (y). Step 2: Therefore the outer pair ((x,z)) is required. Step 3: Even in letter-based questions, pay attention to the order.

Step 1: Check the chains formed in the relation. Step 2: ((1,2)) and ((2,3)) require ((1,3)), and ((2,1)) and ((1,2)) require ((2,2)); these are present. Step 3: A missing pair matters only when it is actually required.

Symmetry and transitivity are different relation properties. Symmetry requires a reverse pair: (a,b) in R implies (b,a) in R. Transitivity requires that whenever (a,b) and (b,c) are in R, the pair (a,c) must also be in R. One property does not generally imply the other. For example, on {1,2}, R = {(1,2),(2,1)} is symmetric, but it is not transitive because (1,2) and (2,1) would require (1,1), which is absent. The identity relation is transitive, the empty relation is vacuously transitive, and the universal relation is transitive because all possible pairs are present. Therefore option A is the false statement.

Step 1: Since ((1,2)) and ((2,1)) are both present, reverse pairs exist. Step 2: But they require ((1,1)) and ((2,2)), which are missing. Step 3: Symmetry does not guarantee transitivity.

Step 1: From ((1,3)) and ((3,5)), ((1,5)) is required. Step 2: Requirements involving ((3,3)) are also satisfied by existing pairs. Step 3: Do not ignore chains involving self-pairs.

Step 1: Combining ((1,3)) and ((3,5)) requires ((1,5)). Step 2: ((1,5)) is not in the relation, so transitivity fails. Step 3: The first job is to identify the common middle element.

Step 1: The main chain is from (a) to (b) and from (b) to (c). Step 2: Transitivity requires the pair from (a) to (c), and it is present. Step 3: Do not reverse the order of the outer elements, even with letters.

Step 1: In ((4,4)) and ((4,7)), the middle element is (4). Step 2: The required outer pair is ((4,7)), which is already present. Step 3: Sometimes the required pair is the same pair already given.

Step 1: From ((7,4)) and ((4,7)), ((7,7)) is required. Step 2: ((7,7)) is not present, so the condition fails. Step 3: When reverse pairs exist, check both self-pairs.

Step 1: The common middle element in the two pairs is (3). Step 2: The outer elements are (2) and (8), so ((2,8)) is required. Step 3: The first and second positions in an ordered pair are very important.

Step 1: From ((1,2)) and ((2,3)), ((1,3)) is required. Step 2: Requirements involving ((2,2)) and ((3,3)) are also satisfied by existing pairs. Step 3: Self-pairs can support the relation, but still check the chains.

Step 1: From ((2,3)) and ((3,2)), ((2,2)) is required. Step 2: ((2,2)) is not present, so transitivity fails. Step 3: An extra reverse pair can create new requirements.

Step 1: First combine ((p,q)) and ((q,r)). Step 2: This gives ((p,r)). Step 3: In a longer chain, start with two pairs at a time.

Step 1: In ((p,r)) and ((r,s)), the middle element is (r). Step 2: By transitivity, the outer pair ((p,s)) must be present. Step 3: In a long chain, a newly obtained pair can be used further.

Step 1: ((1,2)) with ((2,3)) requires ((1,3)), and ((2,3)) with ((3,4)) requires ((2,4)). Step 2: ((1,3)) with ((3,4)) also requires ((1,4)), which is present. Step 3: In step-like relations, check both short and long jumps.

Step 1: Transitivity depends on chains of ordered pairs. Step 2: So for every possible ((a,b)), ((b,c)), check whether ((a,c)) is in the relation. Step 3: In exams, making a small list is the safest method.