Types of relations

Correct answer: A. reflexivity

Step 1: A non-empty set has at least one element. Step 2: Reflexivity needs its diagonal pair, but the empty relation has no pair. Step 3: Hence the empty relation is not reflexive.

Correct answer: A. reflexivity

Step 1: Diagonal pairs are of the form ((a,a)). Step 2: If all diagonal pairs are present, every element is related to itself. Step 3: This is exactly the sign of a reflexive relation.

Correct answer: A. irreflexivity

Step 1: In an irreflexive relation, no element is related to itself. Step 2: Therefore no pair of the form ((a,a)) should be present. Step 3: Absence of diagonal pairs is the key sign of irreflexivity.

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

Step 1: In symmetry, reversing a pair keeps it in the relation. Step 2: The reverse of ((4,7)) is ((7,4)). Step 3: Hence ((7,4)) must be present.

Correct answer: A. ((2,9))

Step 1: Transitivity gives ((a,c)) from ((a,b)) and ((b,c)). Step 2: Here the middle element is (5). Step 3: Therefore ((2,9)) must be in the relation.

Correct answer: A. both cannot occur together

Step 1: Antisymmetry prevents a two-way relation between distinct elements. Step 2: If (a\ne b), then ((a,b)) and ((b,a)) cannot both occur. Step 3: This is different from symmetry, so do not mix the two.

Correct answer: A. equivalence classes

Step 1: An equivalence relation groups similar elements together. Step 2: These groups are called equivalence classes. Step 3: Such classes divide the set into disjoint parts.

Correct answer: A. antisymmetry

Step 1: A partial order relation has reflexivity and transitivity. Step 2: Instead of symmetry, it requires antisymmetry. Step 3: Thus antisymmetry is important for identifying a partial order.

Correct answer: A. equivalence relation

Step 1: All diagonal pairs are present, so it is reflexive. Step 2: ((2,3)) appears with ((3,2)), so it is symmetric. Step 3: The required pairs within ({2,3}) are present, so transitivity holds.

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

Step 1: A reflexive relation must contain the diagonal pair of every element of (A). Step 2: For element (3), ((3,3)) is missing. Step 3: Missing even one diagonal pair breaks reflexivity.

Correct answer: A. symmetry

Step 1: ((1,2)) and ((2,1)) are reverses of each other. Step 2: The reverse of ((3,3)) is itself. Step 3: Every present pair has its reverse, so the relation is symmetric.

Correct answer: A. transitivity

Step 1: ((1,2)) and ((2,3)) require ((1,3)). Step 2: ((1,3)) is present in the relation. Step 3: ((3,3)) creates no missing new pair, so the relation is transitive.

Correct answer: A. irreflexivity

Step 1: Irreflexivity requires no diagonal pair. Step 2: (R) contains neither ((1,1)) nor ((2,2)). Step 3: Hence it is irreflexive, even though it is not symmetric.

Correct answer: A. yes

Step 1: Antisymmetry only forbids two-way pairs between distinct elements. Step 2: Here ((1,2)) is present, but ((2,1)) is absent. Step 3: Diagonal pairs do not violate antisymmetry, so the relation is antisymmetric.

Correct answer: A. because both directions occur while (1\ne2)

Step 1: In antisymmetry, both directions must not occur for distinct elements. Step 2: Here ((1,2)) and ((2,1)) both occur, and (1\ne2). Step 3: Therefore antisymmetry fails.

Correct answer: A. equivalence relation

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

Correct answer: A. because (1\mid2) but (2\nmid1)

Step 1: Symmetry requires the reverse pair to also be valid. Step 2: (1\mid2) is true, but (2\nmid1) is not true. Step 3: Therefore the divisibility relation is not symmetric.

Correct answer: A. because equality is reflexive, symmetric and transitive

Step 1: Every number is equal to itself, so reflexivity holds. Step 2: If (a=b), then (b=a), so symmetry holds. Step 3: If (a=b) and (b=c), then (a=c), so transitivity holds.

Correct answer: A. irreflexive

Step 1: (a\ne a) is never true. Step 2: Therefore ((a,a)) is not in the relation for any (a). Step 3: This is exactly the identity of an irreflexive relation.

Correct answer: A. yes

Step 1: If (a\ne b), then (b\ne a) is also true. Step 2: So the reverse of any related pair is also related. Step 3: Hence the relation is symmetric.

Correct answer: A. because (1R2) and (2R1) hold, but (1R1) does not

Step 1: Transitivity needs ((a,c)) from ((a,b)) and ((b,c)). Step 2: (1\ne2) and (2\ne1) are true, but (1\ne1) is false. Step 3: Hence the relation is not transitive.

Correct answer: A. reflexivity

Step 1: All three diagonal pairs of (A) are in the relation. Step 2: These are ((1,1),(2,2),(3,3)). Step 3: Therefore the relation is reflexive, even with some extra pairs.

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

Step 1: Symmetry requires the reverse of every pair. Step 2: ((1,3)) is present, but ((3,1)) 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)) require ((1,3)). Step 2: The given relation does not contain ((1,3)). Step 3: Hence transitivity fails.

Correct answer: A. (3)

Step 1: A reflexive relation must contain each element's diagonal pair. Step 2: For (3) elements, there are (3) diagonal pairs. Step 3: Hence the minimum number of pairs is (3).