Types of relations

Correct answer: A. universal relation

Step 1: (A\times A) contains all possible ordered pairs. Step 2: If a relation contains all these pairs, it is called the universal relation. Step 3: In such questions, first check whether the relation is the whole Cartesian product.

Correct answer: C. empty relation

Step 1: If a relation contains no ordered pair, it is empty. Step 2: Such a relation is called the empty relation. Step 3: Do not confuse the empty relation with the universal relation, which contains all pairs.

Correct answer: B. identity relation

Step 1: Pairs of the form ((a,a)) are diagonal pairs. Step 2: A relation containing only such required diagonal pairs is the identity relation. Step 3: The identity relation does not contain pairs between distinct elements.

Correct answer: A. when ((a,a)\in R) for every (a\in A)

Step 1: Reflexivity is about every element being related to itself. Step 2: So ((a,a)) must be in the relation for every (a\in A). Step 3: While checking, always look for all diagonal pairs.

Correct answer: A. if ((a,b)\in R), then ((b,a)\in R)

Step 1: In a symmetric relation, reversing the order keeps the pair in the relation. Step 2: Therefore, if ((a,b)) is present, ((b,a)) must also be present. Step 3: In exams, check the reverse of each off-diagonal pair.

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

Step 1: Transitivity uses two connected pairs to require a third pair. Step 2: If ((a,b)) and ((b,c)) are present, ((a,c)) must be present. Step 3: When the middle element matches, check the pair from the first to the third element.

Correct answer: A. reflexive, symmetric and transitive

Step 1: An equivalence relation is a combination of three properties. Step 2: These properties are reflexive, symmetric and transitive. Step 3: If even one of these fails, the relation is not an equivalence relation.

Correct answer: B. reflexive, antisymmetric and transitive

Step 1: A partial order relation gives a structure of comparison. Step 2: It must be reflexive, antisymmetric and transitive. Step 3: Remember the difference: equivalence uses symmetry, while partial order uses antisymmetry.

Correct answer: A. Symmetric and irreflexive

The correct answer is A: symmetric and irreflexive. Symmetry means that whenever (a,b) is in R, (b,a) must also be in R. Here (1,2) and its reverse (2,1) are both included, so the symmetry condition holds. Irreflexive means that no element is related to itself. On A, a reflexive relation would have to contain (1,1), (2,2), and (3,3), but none of these diagonal pairs appears. Therefore R is not reflexive; it is irreflexive. B is wrong because the diagonal pairs are missing. C is wrong because R is not reflexive and it is symmetric. D is wrong because the reverse pair is present. A quick check is to inspect diagonal pairs for reflexivity and reversed off-diagonal pairs for symmetry; never confuse symmetry with reflexivity.

Correct answer: C. universal relation

Step 1: (A\times A) contains all possible ordered pairs of (A). Step 2: If (R=A\times A), the relation contains every pair. Step 3: Hence it is the universal relation.

Correct answer: B. empty relation

Step 1: (\varnothing) contains no ordered pair. Step 2: Therefore, (R) has no pair at all. Step 3: Such a relation is called the empty relation.

Correct answer: A. ((1,1),(2,2),(3,3))

Step 1: In a reflexive relation, each element is related to itself. Step 2: For (1,2,3), the compulsory pairs are ((1,1),(2,2),(3,3)). Step 3: Other pairs may or may not be present, but these diagonal pairs are required.

Correct answer: A. ((3,2))

Step 1: In symmetry, the reverse of a pair must also belong to the relation. Step 2: The reverse of ((2,3)) is ((3,2)). Step 3: Hence ((3,2)) must be present.

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

Step 1: In ((1,2)) and ((2,4)), the middle element (2) matches. Step 2: Transitivity connects the first element to the third element. Step 3: Therefore ((1,4)) must be present.

Correct answer: A. universal relation

Step 1: For (A={1,2}), (A\times A) has exactly these four pairs. Step 2: The relation (R) contains all four pairs. Step 3: Hence (R) is the universal relation.

Correct answer: A. reflexive

Step 1: Reflexivity requires all diagonal pairs. Step 2: ((1,1),(2,2),(3,3)) are present in the relation. Step 3: Therefore it is reflexive, even though it has an extra pair ((1,2)).

Correct answer: B. symmetric

Step 1: ((1,2)) appears with ((2,1)). Step 2: ((2,3)) appears with ((3,2)). Step 3: Every listed pair has its reverse, so the relation is symmetric.

Correct answer: C. transitive

Step 1: ((1,2)) and ((2,3)) are in the relation. Step 2: Transitivity requires ((1,3)), which is present. Step 3: Therefore the relation is transitive.

Correct answer: A. because all diagonal pairs are not present

Step 1: Reflexivity requires ((1,1),(2,2),(3,3)). Step 2: These diagonal pairs are absent in the given relation. Step 3: Hence the relation is not reflexive.

Correct answer: A. because ((2,1)) and ((3,2)) are absent

Step 1: Symmetry requires the reverse of every pair. Step 2: The reverse ((2,1)) of ((1,2)) and the reverse ((3,2)) of ((2,3)) are absent. Step 3: Therefore the relation is not symmetric.

Correct answer: A. because ((1,3)) is absent

Step 1: ((1,2)) and ((2,3)) form a connected chain. Step 2: Transitivity requires ((1,3)). Step 3: Since ((1,3)) is absent, the relation is not transitive.

Correct answer: A. identity relation

Step 1: The condition (a=b) allows only pairs of equal elements. Step 2: Such pairs are of the form ((a,a)). Step 3: Therefore this is the identity relation.

Correct answer: A. equivalence relation

Step 1: (a-a=0) is divisible by (2), so it is reflexive. Step 2: If (a-b) is even, then (b-a) is also even. Step 3: The sum of even differences is even, so it is an equivalence relation.

Correct answer: A. partial order relation

Step 1: For every (a), (a\le a), so it is reflexive. Step 2: If (a\le b) and (b\le a), then (a=b), so it is antisymmetric. Step 3: If (a\le b) and (b\le c), then (a\le c), so it is a partial order.

Correct answer: A. yes

Step 1: All diagonal pairs are present, so it is reflexive. Step 2: The reverse of each diagonal pair is the same pair, so it is symmetric. Step 3: Transitivity also holds for diagonal pairs, so it is an equivalence relation.