Types of relations

Correct answer: A. Transitivity

Step 1: All self-pairs are present, so reflexivity holds. Step 2: Every listed pair has its reverse, so symmetry holds. Step 3: ((1,2)) and ((2,3)) are present but ((1,3)) is missing, so transitivity fails.

Correct answer: A. Equivalence relation

Step 1: (a-a=0) is divisible by (5), so the relation is reflexive. Step 2: If (a-b) is divisible by (5), then (b-a) is also divisible by (5), so it is symmetric. Step 3: The sum of two such differences is also divisible by (5), so transitivity holds.

Correct answer: A. Symmetric but neither reflexive nor transitive

Step 1: (a+a=6) is not true for every (a), so the relation is not reflexive. Step 2: If (a+b=6), then (b+a=6), so it is symmetric. Step 3: ((1,5)) and ((5,1)) exist but ((1,1)) does not, so transitivity fails.

Correct answer: A. Symmetric but neither reflexive nor transitive

Step 1: (a-a=0) is rational, so the relation is not reflexive. Step 2: If (a-b) is irrational, then (b-a) is also irrational, so it is symmetric. Step 3: Differences like (\sqrt{2}) and (-\sqrt{2}) can add to (0), so transitivity need not hold.

Correct answer: A. Symmetric and transitive but not reflexive

Step 1: Self-pairs for (1) and (3) will not occur, so the relation is not reflexive. Step 2: The condition that both elements are even remains true after reversing order, so it is symmetric. Step 3: Chains stay within even elements, so transitivity holds.

Correct answer: A. ({1,4},{2,5},{3,6})

Step 1: Elements with the same remainder on division by (3) are grouped together. Step 2: (1,4) have remainder (1), (2,5) have remainder (2), and (3,6) have remainder (0). Step 3: Equivalence classes divide the set into non-overlapping parts.

Correct answer: A. (2^{20})

Step 1: (A\times A) has (5^2=25) ordered pairs. Step 2: A reflexive relation must contain (5) self-pairs. Step 3: The remaining (20) pairs are optional, so the number is (2^{20}).

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

Step 1: The four self-pairs can be chosen independently. Step 2: There are (\frac{4\cdot3}{2}=6) unordered pairs of distinct elements. Step 3: Total independent choices are (4+6=10), so the number of symmetric relations is (2^{10}).

Correct answer: B. 3^6

Correct answer: B, 3^6 = 729. A reflexive relation on a four-element set must contain all four diagonal pairs, so those pairs contribute no choices. The number of unordered pairs of distinct elements is C(4,2)=6. For each such pair, antisymmetry permits exactly three choices: include neither direction, include the first direction only, or include the reverse direction only. Both directions cannot be included together. Therefore the total number is 3^6 = 729. Option A, 2^4, counts only the four diagonal positions and incorrectly treats required pairs as optional. Option B is correct because it captures three independent choices for each of six unordered pairs. Option C, 2^8, does not reflect the three-way choice and has no valid basis. Option D, 2^12, treats all off-diagonal ordered pairs as independent, which could include opposite pairs and violate antisymmetry. Memory cue: reflexive fixes the diagonal; antisymmetric leaves three choices for each unordered pair of distinct elements.

Correct answer: A. It is both the universal relation and an equivalence relation

Step 1: For a set of three elements, (A\times A) has (9) pairs and all (9) are listed here. Step 2: Therefore, it is the universal relation. Step 3: The universal relation is reflexive, symmetric and transitive, so it is also an equivalence relation.

Correct answer: A. Symmetry

Step 1: For (a=a), (a\le a) and (a-a=0\le2), so reflexivity holds. Step 2: ((1,2)) is in the relation, but ((2,1)) is not. Step 3: Hence symmetry fails; one missing reverse pair is enough.

Correct answer: A. No, because ((1,3)) and ((3,4)) exist but ((1,4)) does not

Step 1: ((1,3)) has difference (2), so it is in the relation. Step 2: ((3,4)) has difference (1), so it is also in the relation. Step 3: ((1,4)) has difference (3), which is beyond the limit, so transitivity fails.

Correct answer: A. Equivalence relation

Step 1: For every (a), (a^3=a^3), so reflexivity holds. Step 2: From (a^3=b^3), we get (b^3=a^3), so symmetry holds. Step 3: If (a^3=b^3) and (b^3=c^3), then (a^3=c^3), so it is transitive too.

Correct answer: A. Reflexive and transitive but not symmetric

Step 1: (a^2\le a^2), so the relation is reflexive. Step 2: If (a^2\le b^2) and (b^2\le c^2), then (a^2\le c^2), so it is transitive. Step 3: (1^2\le2^2) is true but (2^2\le1^2) is false, so it is not symmetric.

Correct answer: A. Symmetric but neither reflexive nor transitive

Step 1: For ((1,1)), the sum is (2), so it is not reflexive. Step 2: The sum stays the same when order is reversed, so the relation is symmetric. Step 3: ((1,3)) and ((3,1)) exist but ((1,1)) does not, so transitivity fails.

Correct answer: A. Reflexive and transitive but not symmetric

Step 1: For every (a), (a\le a), so it is reflexive. Step 2: (a\le b) and (b\le c) imply (a\le c), so it is transitive. Step 3: (1\le2) is true but (2\le1) is false, so it is not symmetric.

Correct answer: A. Transitive but neither reflexive nor symmetric

Step 1: No number is less than itself, so reflexivity fails. Step 2: If (a<b), then (b<a) cannot hold, so symmetry fails. Step 3: From (a<b) and (b<c), we get (a<c), so transitivity holds.

Correct answer: A. It is an equivalence relation

Step 1: All self-pairs are given, so the relation is reflexive. Step 2: Extra pairs such as ((1,2),(2,4),(1,4)) have their reverse pairs, so it is symmetric. Step 3: ({1,2,4}) is a complete closed block and (3) is alone, so the relation is transitive too.

Correct answer: A. Reflexivity

Step 1: A reflexive relation must contain ((4,4)). Step 2: Here ((4,4)) is missing, so reflexivity fails. Step 3: In exams, checking self-pairs first saves a lot of time.

Correct answer: A. Transitive but neither reflexive nor symmetric

Step 1: No self-pair is present, so the relation is not reflexive. Step 2: ((1,2)) exists but ((2,1)) does not, so it is not symmetric. Step 3: ((1,2)) and ((2,3)) require ((1,3)), which is present, so transitivity holds.

Correct answer: A. Symmetric but neither reflexive nor transitive

Step 1: (|a-a|=0), so no self-pair occurs and reflexivity fails. Step 2: (|a-b|=|b-a|), so symmetry holds. Step 3: ((1,3)) and ((3,1)) exist but ((1,1)) does not, so transitivity fails.

Correct answer: A. It is an equivalence relation

Step 1: In the identity relation, every element is related only to itself, so it is reflexive. Step 2: The reverse of ((a,a)) is still ((a,a)), so it is symmetric. Step 3: Chains of self-pairs again give self-pairs, so transitivity holds.

Correct answer: A. It is symmetric and transitive but not reflexive

Step 1: On a non-empty set, reflexivity needs self-pairs, but the empty relation has no pair. Step 2: There is no pair that can violate symmetry or transitivity. Step 3: Hence it is symmetric and transitive, but not reflexive.

Correct answer: A. (4)

Step 1: A reflexive relation must contain ((a,a)) for every element. Step 2: For four elements, ((1,1),(2,2),(3,3),(4,4)) are necessary. Step 3: Hence a minimum of (4) pairs must be added.

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

Step 1: Symmetry requires ((b,a)) for every ((a,b)). Step 2: ((1,2)) and ((2,1)) are already both present. Step 3: The reverse of ((2,3)), namely ((3,2)), is missing, so it must be added.