Types of relations

Correct answer: A. ((5,2))

Step 1: An equivalence relation is symmetric. Step 2: Because of symmetry, ((2,5)) implies ((5,2)). Step 3: In equivalence relation questions, first write the reverse pair of a given pair.

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

Step 1: An equivalence relation is transitive. Step 2: From ((1,2)) and ((2,4)), ((1,4)) must be present. Step 3: Symmetry can then give ((4,1)), but the direct transitive conclusion is ((1,4)).

Correct answer: A. (a=b)

Step 1: A partial order relation is antisymmetric. Step 2: In antisymmetry, if both reverse pairs are present, the elements must be equal. Step 3: This conclusion is very useful in partial order questions.

Correct answer: A. It is always reflexive, symmetric and transitive

Step 1: An identity relation contains only self-pairs. Step 2: These pairs give reflexivity, are their own reverses, and do not break transitivity. Step 3: So identity relation is a simple example of an equivalence relation.

Correct answer: A. It is always reflexive and symmetric

Step 1: The universal relation contains all pairs of (A\times A). Step 2: Hence every self-pair and every reverse pair is present. Step 3: Think of the universal relation as the complete collection of all pairs.

Correct answer: C. (2^{16})

Step 1: (A\times A) has (4^2=16) pairs. Step 2: Every relation is a subset of (A\times A). Step 3: The number of subsets of (16) pairs is (2^{16}).

Correct answer: B. (2^{10})

Step 1: The number of symmetric relations is (2^{\frac{n(n+1)}{2}}). Step 2: For (n=4), (\frac{4\cdot5}{2}=10) independent choices. Step 3: Therefore the number is (2^{10}).

Correct answer: A. 960

Correct answer: A, 960. For four elements, there are four diagonal pairs and C(4,2)=6 unordered pairs of distinct elements. In a symmetric relation, each diagonal pair can independently be included or omitted, giving 2^4 choices. For each unordered distinct pair, symmetry forces both directions to be included together or both to be omitted, giving 2 choices; hence the off-diagonal part gives 2^6 choices. Thus there are 2^4 × 2^6 = 2^10 symmetric relations in total. To be reflexive, all four diagonal pairs must be present. Once they are fixed, the six unordered pairs remain free, so there are 2^6 reflexive symmetric relations. Therefore the number that are symmetric but not reflexive is 2^10 − 2^6 = 1024 − 64 = 960. Option A is correct. Option B counts only the reflexive symmetric relations. Option C counts all symmetric relations, including reflexive ones. Option D does not represent the required independent choices. Warning: “not reflexive” means at least one diagonal pair is absent, not that all diagonal pairs must be absent.

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

Step 1: Reflexivity fixes all (n) self-pairs. Step 2: For symmetry, only non-self reverse-pair groups remain independent. Step 3: There are (\frac{n(n-1)}{2}) such groups, so the count is (2^{\frac{n(n-1)}{2}}).

Correct answer: B. No, symmetry gives only reverse pairs

Step 1: Symmetry gives only ((2,1)) from ((1,2)) and ((3,2)) from ((2,3)). Step 2: ((1,3)) would be guaranteed by transitivity, not symmetry. Step 3: Do not mix properties; each property has a separate role.

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

Step 1: From ((1,2)) and ((2,1)), transitivity requires ((1,1)). Step 2: From ((2,1)) and ((1,2)), it requires ((2,2)). Step 3: A pair of reverse pairs can force self-pairs through transitivity.

Correct answer: B. No, transitivity is also needed

Step 1: An equivalence relation needs three properties. Step 2: Along with reflexivity and symmetry, transitivity is also necessary. Step 3: Do not declare equivalence after checking only two properties.

Correct answer: B. No, symmetry is also needed

Step 1: Equivalence relation needs symmetry along with reflexivity and transitivity. Step 2: Many relations like (\le) are reflexive and transitive but not symmetric. Step 3: All three properties are required for equivalence.

Correct answer: B. No, self-pairs for all elements need not occur

Step 1: Symmetry and transitivity can create self-pairs for connected elements. Step 2: But elements not related to anything may still miss their self-pairs. Step 3: Reflexivity requires checking self-pairs for every element separately.

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

Step 1: Reflexivity requires self-pairs for (1,2,3). Step 2: The relation has ((1,1)) and ((2,2)), but not ((3,3)). Step 3: If even one element lacks its self-pair, the relation is not reflexive.

Correct answer: A. It is always symmetric and transitive

Step 1: In the empty relation, no pair exists to violate symmetry. Step 2: There are no linked pairs to violate transitivity either. Step 3: But on a non-empty set it is not reflexive because self-pairs are missing.

Correct answer: A. It is considered reflexive, symmetric and transitive

Step 1: In the empty set, there is no element that can violate reflexivity. Step 2: There are no pairs that can violate symmetry or transitivity. Step 3: In empty-set questions, conditions can be vacuously true.

Correct answer: A. Identity relation

Step 1: An identity relation contains only self-pairs. Step 2: Self-pairs satisfy symmetry and do not violate antisymmetry. Step 3: Reverse pairs of unequal elements would break antisymmetry.

Correct answer: A. No pair can occur between unequal elements

Step 1: Symmetry says if ((a,b)) is present, then ((b,a)) must be present. Step 2: Antisymmetry does not allow both pairs for unequal (a,b). Step 3: Therefore no pair between unequal elements can remain.

Correct answer: B. No, because ((1,3)) is missing

Step 1: ((1,2)) and ((2,3)) are in the relation. Step 2: Transitivity requires ((1,3)), which is missing. Step 3: Even with self-pairs present, transitivity can fail separately.

Correct answer: A. Yes

Step 1: All self-pairs are present, so it is reflexive. Step 2: No reverse pair for unequal elements occurs, so it is antisymmetric. Step 3: ((1,2)) and ((2,3)) require ((1,3)), which is present, so it is transitive.

Correct answer: A. Because ((1,2)) and ((2,1)) both exist while (1\ne2)

Step 1: A partial order requires antisymmetry. Step 2: Here for unequal (1) and (2), both reverse pairs are present. Step 3: Once antisymmetry fails, the relation cannot be a partial order.

Correct answer: A. (\le) on real numbers

Step 1: (\le) is reflexive, antisymmetric and transitive, so it is a partial order. Step 2: It is not symmetric because (2\le3) but (3\le2) is false. Step 3: Symmetry is required for equivalence relation.

Correct answer: A. Same remainder on division by (2) over integers

Step 1: The same-remainder relation is reflexive, symmetric and transitive, so it is an equivalence relation. Step 2: It is not antisymmetric because different numbers can have the same remainder. Step 3: Distinguish symmetric equivalence relations from partial orders.

Correct answer: A. First self-pairs, then reverse pairs, then linked pairs

Step 1: Self-pairs help check reflexivity quickly. Step 2: Reverse pairs help check symmetry or antisymmetry. Step 3: Linked pairs help check transitivity and then decide the final type.