Types of relations

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

The correct answer is A. A relation is symmetric when every pair has its reverse: (a,b)∈R must imply (b,a)∈R. Option A contains both (1,2) and (2,1), so it is symmetric. It is not reflexive because a reflexive relation on A must contain all three diagonal pairs (1,1), (2,2), and (3,3), and none is present. Option B is reflexive, although it is also symmetric, so it does not satisfy “not reflexive.” Option C is not symmetric because (1,2) is present but (2,1) is absent. Option D contains all diagonal pairs and the reverse pair, so it is both reflexive and symmetric. The useful distinction is that symmetry checks reversal, whereas reflexivity checks self-pairs; one property does not automatically imply the other.

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

Step 1: The universal relation contains every pair from (A\times A). Step 2: In ((2,3)), both elements belong to (A). Step 3: So this pair must be present in the universal relation.

Correct answer: A. reflexivity

Step 1: Being related to itself means the pair ((a,a)) is present. Step 2: If this happens for every element, the relation is reflexive. Step 3: In such questions, check all diagonal pairs.

Correct answer: A. irreflexivity

Step 1: Not being related to itself means diagonal pairs are absent. Step 2: If ((a,a)) is absent for every (a), the relation is irreflexive. Step 3: Off-diagonal pairs can still be present in an irreflexive relation.

Correct answer: A. ((8,5))

Step 1: Symmetry requires the reverse of every pair. Step 2: The reverse of ((5,8)) is ((8,5)). Step 3: Hence ((8,5)) must be in the relation.

Correct answer: A. ((3,6))

Step 1: Transitivity needs ((a,c)) from ((a,b)) and ((b,c)). Step 2: Here the middle element is (4). Step 3: Therefore ((3,6)) is necessary.

Correct answer: A. antisymmetry

Step 1: An equivalence relation needs reflexivity, symmetry and transitivity. Step 2: Antisymmetry is linked with partial order relations. Step 3: Therefore antisymmetry is not required for equivalence.

Correct answer: A. symmetry

Step 1: A partial order requires reflexivity, antisymmetry and transitivity. Step 2: Symmetry is not a required condition for partial order. Step 3: Therefore symmetry is the wrongly added property here.

Correct answer: A. reflexivity

Step 1: Reflexivity requires all diagonal pairs. Step 2: ((1,1),(2,2),(3,3)) are present. Step 3: Hence the relation is reflexive, even with the extra pair ((2,1)).

Correct answer: A. symmetry

Step 1: ((1,2)) is accompanied by ((2,1)). Step 2: ((3,1)) is accompanied by ((1,3)). Step 3: Every pair has its reverse, so the relation is symmetric.

Correct answer: A. transitivity

Step 1: ((1,2)) and ((2,4)) require ((1,4)). Step 2: ((1,4)) is present in the relation. Step 3: Therefore this relation is transitive.

Correct answer: A. irreflexivity

Step 1: Irreflexivity requires no diagonal pair. Step 2: The relation has none of ((1,1),(2,2),(3,3)). Step 3: Hence the relation is irreflexive.

Correct answer: A. because ((3,2)) is absent

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

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

Step 1: ((1,2)) and ((2,3)) are present. Step 2: Transitivity requires ((1,3)). Step 3: Since ((1,3)) is absent, the relation is not transitive.

Correct answer: A. no

Step 1: An equivalence relation needs all three properties. Step 2: Here transitivity is missing. Step 3: Therefore it is not an equivalence relation.

Correct answer: A. no

Step 1: A partial order requires reflexivity, antisymmetry and transitivity. Step 2: Here antisymmetry is missing. Step 3: Therefore it is not a partial order relation.

Correct answer: A. (9)

Step 1: (A\times A) contains all ordered pairs. Step 2: Since (A) has (3) elements, there are (3\times3=9) pairs. Step 3: This is the number of pairs in the universal relation.

Correct answer: A. (6)

Step 1: In the identity relation, each element contributes one diagonal pair. Step 2: With (6) elements, there are (6) diagonal pairs. Step 3: Hence the identity relation has (6) pairs.

Correct answer: D. (36)

Step 1: The universal relation is (A\times A). Step 2: For (6) elements, (A\times A) has (6^2=36) pairs. Step 3: Therefore the universal relation contains (36) pairs.

Correct answer: A. symmetry

Step 1: To fail symmetry, a pair must exist without its reverse. Step 2: The empty relation has no pair at all. Step 3: Therefore the empty relation is considered symmetric.

Correct answer: A. transitivity

Step 1: To fail transitivity, two connected pairs are needed. Step 2: The empty relation has no pairs, so such a situation never occurs. Step 3: Hence the empty relation is considered transitive.

Correct answer: A. reflexive, symmetric and transitive

Step 1: The identity relation has all diagonal pairs, so it is reflexive. Step 2: The reverse of a diagonal pair is itself, so it is symmetric. Step 3: Transitivity is also satisfied by diagonal pairs.

Correct answer: A. reflexive, symmetric and transitive

Step 1: The universal relation has all pairs, so it has all diagonal pairs. Step 2: The reverse of every pair is also present. Step 3: The pair needed for transitivity is also present.

Correct answer: A. reflexive and transitive

Step 1: Since ((1,1)) and ((2,2)) are present, the relation is reflexive. Step 2: ((1,2)) does not create a missing transitivity requirement. Step 3: But ((2,1)) is absent, so it is not symmetric.

Correct answer: A. symmetry

Step 1: ((1,2)) is present along with ((2,1)). Step 2: The two pairs are reverses of each other. Step 3: Therefore the relation is symmetric.