Transitive relation

Step 1: ((1,2)) and ((2,1)) require ((1,1)), which is present. Step 2: ((2,1)) and ((1,2)) require ((2,2)), which is also present. Step 3: Reverse pairs can still be transitive when the needed diagonal pairs are included.

Step 1: ((1,3)) and ((3,4)) are in the relation. Step 2: Therefore ((1,4)) must be present, but it is missing. Step 3: Longer chains formed inside the relation also create transitive requirements.

Step 1: To apply transitivity, a second pair of the form ((b,c)) is needed. Step 2: If no such pair exists, then ((a,b)) alone does not force a new pair. Step 3: In transitivity questions, first identify whether a chain is actually formed.

Step 1: If (|a|=|b|) and (|b|=|c|), then equality gives (|a|=|c|). Step 2: Hence ((a,c)) also belongs to the relation. Step 3: Relations based on equality of a value are often transitive.

Step 1: If (a) and (b) have the same remainder, and (b) and (c) also have the same remainder, then (a) and (c) have the same remainder. Step 2: So ((a,c)) belongs to the relation. Step 3: Same-remainder relations are strong examples of transitivity.

Step 1: ((1,3)) and ((3,2)) require ((1,2)). Step 2: ((1,2)) is present, so the main transitive condition is satisfied. Step 3: Identify the common middle element and match the required pair.

Step 1: The common middle element in ((1,3)) and ((3,2)) is (3). Step 2: Transitivity requires ((1,2)). Step 3: The required pair is formed from the first element of the first pair and the second element of the second pair.

Step 1: In ((5,5)) and ((5,8)), the common middle element is (5). Step 2: Transitivity gives ((5,8)), which is already present. Step 3: Sometimes the transitive rule gives an old pair, not a new one.

Step 1: ((2,3)) and ((3,4)) require ((2,4)), which is present. Step 2: Chains involving ((2,2)) give pairs such as ((2,2)) or ((2,3)), which are already present. Step 3: Check each possible chain separately.

Step 1: The condition of transitivity is based on two connected pairs. Step 2: If both pairs are present but the required ((a,c)) is absent, the relation is not transitive. Step 3: When writing a counterexample, clearly state the status of all three pairs.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.