Transitive relation

Correct answer: A. Yes, because every element is related to itself

Step 1: In transitivity, if ((a,b)\in R) and ((b,c)\in R), then ((a,c)\in R) must also be present. Step 2: Here only self-pairs are present, so the condition is never violated. Step 3: In exams, remember that the identity relation is transitive.

Correct answer: A. It is transitive

Step 1: Since ((1,2)) and ((2,3)) are present, ((1,3)) must be present. Step 2: ((1,3)) is already in the relation, so the key transitivity condition is satisfied. Step 3: In such questions, match the middle element carefully.

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

Step 1: From ((1,2)) and ((2,3)), the pair ((1,3)) is required. Step 2: The relation does not contain ((1,3)), so the condition fails. Step 3: For transitivity, check the first and third elements, not the reverse pair.

Correct answer: A. If ((a,b)\in R) and ((b,c)\in R), then ((a,c)\in R)

Step 1: Transitivity checks a chain of two related pairs. Step 2: If one pair takes (a) to (b) and another takes (b) to (c), then (a) must relate to (c). Step 3: Keep this separate from symmetry and reflexivity.

Correct answer: A. Yes, because no violating case occurs

Step 1: Transitivity is checked only when both ((a,b)) and ((b,c)) are present. Step 2: In an empty relation, no such pair chain exists, so the condition is not violated. Step 3: The empty relation is commonly considered transitive in exams.

Correct answer: A. It is transitive

Step 1: A universal relation contains every possible ordered pair from (A). Step 2: So whenever ((a,b)) and ((b,c)) are present, ((a,c)) is also present. Step 3: Remember that a universal relation is transitive.

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

Step 1: In ((2,4)) and ((4,6)), the middle element (4) matches. Step 2: Therefore the pair from the first element (2) to the last element (6), namely ((2,6)), is required. Step 3: Remove the middle element and form the ordered pair of the outer elements.

Correct answer: A. It is transitive

Step 1: Check all possible chains in the relation. Step 2: From ((1,2)) and ((2,2)), ((1,2)) is required and it is present; self-pairs do not break the condition. Step 3: Check each possible chain carefully.

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

Step 1: From ((1,2)) and ((2,1)), the pair ((1,1)) is required. Step 2: From ((2,1)) and ((1,2)), the pair ((2,2)) is required. Step 3: When reverse pairs appear together, check the self-pairs.

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

Step 1: For transitivity, ((1,2)) and ((2,3)) require ((1,3)). Step 2: The first option contains all three required pairs. Step 3: Before choosing, identify the required pair in each option.

Correct answer: A. Transitive

Step 1: From ((3,4)) and ((4,4)), ((3,4)) is required. Step 2: From ((3,3)) and ((3,4)), ((3,4)) is also required and present. Step 3: Include cases involving self-pairs while checking.

Correct answer: A. ((5,9))

Step 1: The second element of the first pair and the first element of the second pair are both (7). Step 2: By transitivity, ((5,9)) must be present. Step 3: Do not reverse the order in an ordered pair.

Correct answer: A. (R) is not transitive

Step 1: The main condition of transitivity requires ((a,c)\in R). Step 2: The question states that ((a,c)\notin R), so the condition fails. Step 3: One counterexample is enough to show non-transitivity.

Correct answer: A. Yes

Step 1: The main chain is ((1,2)) and ((2,3)), which requires ((1,3)). Step 2: ((1,3)) is present, and ((3,3)) creates no missing required pair. Step 3: An extra self-pair does not make a relation non-transitive.

Correct answer: A. Identity relation

Step 1: In an identity relation, each element is related only to itself. Step 2: Any chain formed in it keeps the needed self-pair inside the relation. Step 3: Remember the identity relation as a standard example of a transitive relation.

Correct answer: A. It is transitive

Step 1: From ((1,2)) and ((2,3)), ((1,3)) is required. Step 2: ((1,3)) is present, and the requirements involving ((2,2)) are also satisfied. Step 3: Checking each possible chain is the safest method.

Correct answer: A. ((1,2))

Step 1: In ((1,3)) and ((3,2)), the middle element is (3). Step 2: Transitivity requires ((1,2)), but it is not in the relation. Step 3: As soon as a chain appears, write the required outer pair.

Correct answer: A. Finding pairs where the second element of the first pair equals the first element of the second pair

Step 1: Transitivity is based on a chain of two pairs. Step 2: A chain is formed when the second element of one pair equals the first element of another pair. Step 3: Identifying the correct chain matters more than counting pairs.

Correct answer: A. Transitive

Step 1: Transitivity requires a chain of two pairs. Step 2: A single pair ((2,5)) does not form a pair chain of the required type. Step 3: If no violating chain exists, the relation can be considered transitive.

Correct answer: A. Yes

Step 1: From ((1,2)) and ((2,1)), ((1,1)) is required. Step 2: From ((2,1)) and ((1,2)), ((2,2)) is required; both are present. Step 3: With reverse pairs, always check the self-pairs.

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

Step 1: Combining ((2,1)) and ((1,2)) requires ((2,2)). Step 2: ((2,2)) is not present in the relation, so transitivity fails. Step 3: One self-pair may not be enough when reverse pairs exist.

Correct answer: A. The required ((1,4)) from ((1,2)) and ((2,4)) is present

Step 1: The key chain is from ((1,2)) to ((2,4)). Step 2: This requires ((1,4)), which is present. Step 3: Wrong options often show reverse pairs, so be careful.

Correct answer: A. ((a,d)\in R)

Step 1: The middle element (b) connects the two pairs. Step 2: By transitivity, the pair from (a) to (d) must be in (R). Step 3: Apply the same rule to letters as you do to numbers.

Correct answer: A. Yes

Step 1: From ((1,1)) and ((1,2)), ((1,2)) is required and present. Step 2: From ((1,2)) and ((2,3)), ((1,3)) is required and present. Step 3: List all possible chains and check them.

Correct answer: A. ((1,4))

Step 1: Combining ((1,3)) and ((3,4)) requires ((1,4)). Step 2: ((1,4)) is missing, so the relation is not transitive. Step 3: In a longer list, also check chains formed by later pairs.