Transitive relation

Correct answer: A. (R={(1,2),(2,3)})

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.

Correct answer: A. (R) can be considered transitive

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.

Correct answer: A. (R) is transitive

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.

Correct answer: A. ((2,6))

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.

Correct answer: A. Yes

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.

Correct answer: A. Yes

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.

Correct answer: A. ((1,3))

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.

Correct answer: A. Transitive

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.

Correct answer: A. A required pair like ((2,1)) is missing

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.

Correct answer: A. ((x,z))

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.

Correct answer: A. ((3,1)) is not required, and no chain starts from (3); actually it is transitive

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.

Correct answer: A. Every symmetric relation is necessarily transitive

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.

Correct answer: A. ((1,1)) and ((2,2)) are missing

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.

Correct answer: A. It is transitive

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.

Correct answer: A. ((1,5)) is missing

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.

Correct answer: A. ((a,b)) and ((b,c)) give ((a,c)), which is present

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.

Correct answer: A. ((4,7))

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.

Correct answer: A. ((7,7)) is missing

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.

Correct answer: A. From ((2,3)) and ((3,8)), ((2,8)) is required

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.

Correct answer: A. Transitive

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.

Correct answer: A. ((2,2))

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.

Correct answer: A. ((p,r))

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.

Correct answer: A. ((p,s))

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.

Correct answer: A. It is transitive

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.

Correct answer: A. Make every ((a,b)) and ((b,c)) chain and check ((a,c))

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.