Types of relations

Correct answer: A. Symmetric

Step 1: In a symmetric relation, every pair has its reverse pair. Step 2: Any pair in (R\cup S) comes from either (R) or (S), and its reverse comes with it. Step 3: The union of two symmetric relations remains symmetric.

Correct answer: B. No, not always

Step 1: Transitivity requires a third pair from two linked pairs. Step 2: In a union, one pair may come from one relation and the second from another, so the third pair may be missing. Step 3: Always test transitivity of a union separately.

Correct answer: B. ({(1,2),(2,3),(1,3)})

Step 1: ((1,2)) and ((2,3)) require ((1,3)). Step 2: The second option contains this required pair. Step 3: In transitivity, check only pairs that can be linked through a middle element.

Correct answer: B. No, because ((3,2)) is missing

Step 1: Symmetry depends on reverse pairs, not on the number of pairs. Step 2: The reverse of ((2,3)) is ((3,2)), which is missing. Step 3: Even one non-self pair without its reverse breaks symmetry.

Correct answer: A. Because the reverse of ((2,2)) is itself

Step 1: Symmetry requires the reverse of every pair to be in the relation. Step 2: The reverse of ((2,2)) is again ((2,2)), which is present. Step 3: Self-pairs automatically satisfy symmetry.

Correct answer: C. (4)

Step 1: (A\times A) has (2^2=4) pairs. Step 2: For reflexivity, ((1,1),(2,2)) are fixed, while the other two pairs may be chosen or not. Step 3: Therefore the count is (2^2=4).

Correct answer: B. 2^6

The correct answer is B, 2^6. A relation on A is any subset of A×A. Since A has 3 elements, A×A has 3×3=9 ordered pairs. Three of them are diagonal pairs: (1,1), (2,2), and (3,3). An irreflexive relation cannot contain any diagonal pair, so only the remaining 9−3=6 off-diagonal pairs are available. Each available pair can independently be included or excluded. A set of six independently optional objects has 2^6 subsets, so there are 2^6 irreflexive relations. Option A counts only three choices and misses the six available pairs. Option C allows the forbidden diagonal pairs and therefore counts all relations, not just irreflexive ones. Option D uses the wrong counting structure. Memory cue: for an n-element set, an irreflexive relation has 2^(n^2−n) possibilities, because all n diagonal pairs are forbidden.

Correct answer: C. (2^{n^2})

Step 1: (A\times A) has (n^2) ordered pairs. Step 2: Any relation is a subset of (A\times A). Step 3: The number of subsets of (n^2) pairs is (2^{n^2}).

Correct answer: B. (2^{n^2-n})

Step 1: Total pairs are (n^2). Step 2: Reflexivity fixes (n) self-pairs. Step 3: The remaining (n^2-n) pairs are free, so the count is (2^{n^2-n}).

Correct answer: C. 216 = 2^3 × 3^3

Correct answer: C, 216. An antisymmetric relation cannot contain both (a,b) and (b,a) when a and b are distinct. The three diagonal pairs (a,a) are special: each may be included or omitted, giving 2^3 choices. There are C(3,2)=3 unordered pairs of distinct elements. For each pair, there are exactly three allowed choices: include neither direction, include only the first direction, or include only the reverse direction. Thus the total is 2^3 × 3^3 = 8 × 27 = 216. Option A, 2^6, forgets the three possible choices for each off-diagonal pair. Option B, 2^8, has no valid counting basis and does not account for the three choices correctly. Option C gives the correct product. Option D, 3^3, counts the off-diagonal choices but forgets that each diagonal pair can independently be selected or omitted. Memory cue: for antisymmetry, diagonal pairs have 2 choices, while each unordered distinct pair has 3 choices—not 4, because both opposite directions together are forbidden.

Correct answer: A. (2^{\frac{n(n+1)}{2}})

Step 1: The (n) self-pairs are independent. Step 2: The non-self reverse-pair groups are (\frac{n(n-1)}{2}). Step 3: Total independent choices are (n+\frac{n(n-1)}{2}=\frac{n(n+1)}{2}), so the count is (2^{\frac{n(n+1)}{2}}).

Correct answer: A. Reflexive and symmetric

Step 1: All three self-pairs are present, so reflexivity holds. Step 2: ((1,2)) has ((2,1)), and ((2,3)) has ((3,2)), so symmetry holds. Step 3: Transitivity needs a separate check and should not be assumed immediately.

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

Step 1: ((1,2)) and ((2,3)) are both in the relation. Step 2: Transitivity requires ((1,3)), but it is missing. Step 3: Missing even one required pair destroys transitivity.

Correct answer: A. R is symmetric and irreflexive

Correct answer: A. The empty relation contains no ordered pair at all. A relation is symmetric if whenever (x,y) belongs to it, (y,x) also belongs to it. Since the empty relation has no pair that can violate this condition, it is symmetric vacuously. It is also irreflexive because irreflexivity requires that no pair (x,x) belong to the relation, and the empty relation certainly contains none. Option A is therefore correct. Option B is wrong because A is nonempty, so reflexivity would require every diagonal pair (a,a), but none is present. Option C is wrong because an equivalence relation must be reflexive, symmetric, and transitive; the empty relation fails reflexivity here. Option D is wrong because the empty relation is symmetric, and it is also transitive vacuously: there are no two related pairs creating a condition to check. A useful warning is that “no counterexample exists” makes an implication-type property such as symmetry or transitivity true, but it does not make reflexivity true on a nonempty set.

Correct answer: D. A reflexive relation has no self-pair

Step 1: Reflexive relation is defined by the presence of self-pairs. Step 2: So saying it has no self-pair is always false. Step 3: Identify statements that directly contradict definitions.

Correct answer: A. Reflexive and transitive, but not symmetric

Step 1: All self-pairs are present, so it is reflexive. Step 2: ((1,2)) and ((2,3)) require ((1,3)), which is present, so transitivity is satisfied here. Step 3: ((1,2)) lacks ((2,1)), so it is not symmetric.

Correct answer: C. Universal relation

Step 1: For three elements, (A\times A) has (9) ordered pairs. Step 2: The given relation contains all (9) possible pairs. Step 3: A relation containing all possible pairs is universal.

Correct answer: A. Empty has no pair, universal has all possible pairs

Step 1: The empty relation has no ordered pair. Step 2: The universal relation is (A\times A), so it contains all possible pairs. Step 3: Do not interchange these two definitions in exams.

Correct answer: A. Because it has no self-pairs like ((a,a))

Step 1: A reflexive relation needs the self-pair of every element. Step 2: The empty relation has no pair at all, so it has no self-pairs. Step 3: On a non-empty set, the empty relation is not reflexive.

Correct answer: B. Empty relation is (\varnothing) and identity relation is ({(1,1)})

Step 1: The empty relation has no pair. Step 2: On a one-element set, the identity relation contains ((1,1)). Step 3: Even for a small set, distinguish empty and identity relations.

Correct answer: B. Reflexivity of the whole relation

Step 1: ((a,b)) and ((b,a)) are reverse pairs. Step 2: This shows symmetry for these pairs, but says nothing about self-pairs of all elements. Step 3: Reflexivity must be checked separately using ((x,x)).

Correct answer: B. Like (\ge)

Step 1: The pairs show the first element is greater than or equal to the second. Step 2: For example, (3) is related to (2) and (1), and (2) is related to (1). Step 3: Recognize this pattern as similar to the (\ge) relation.

Correct answer: A. Equivalence relation

Step 1: Every person has the same birthday date as themselves, so reflexivity holds. Step 2: If (a)'s date matches (b)'s date, then (b)'s date matches (a)'s date, so symmetry holds. Step 3: Matching the same date passes through a third person, so it is an equivalence relation.

Correct answer: A. Equivalence relation

Step 1: Every student is in the same class as themselves, so it is reflexive. Step 2: If (a) is in the same class as (b), then (b) is in the same class as (a), so it is symmetric. Step 3: The same-class link transfers to a third student, so it is an equivalence relation.

Correct answer: C. Transitive

Step 1: A person is not taller than themselves, so it is not reflexive. Step 2: If (a) is taller than (b), then (b) is not taller than (a). Step 3: If (a) is taller than (b) and (b) is taller than (c), then (a) is taller than (c), so it is transitive.