Types of relations

Correct answer: A. It is symmetric

Answer: A, it is symmetric. To test symmetry, inspect every pair: (1,2) is accompanied by (2,1), and diagonal pairs (1,1) and (2,2) are their own reverses. Therefore every pair in R has its reverse in R. It is not reflexive on A because (3,3) is missing. It is not antisymmetric because both (1,2) and (2,1) occur even though 1 and 2 are different. It is not universal because A×A would have nine pairs, whereas R has only four and contains no pair involving 3. Option A is therefore the only valid property among the choices. Common confusion: symmetry and antisymmetry are not opposites in ordinary language; check their definitions directly.

Correct answer: B. 4

The correct answer is B, 4. A relation is symmetric if (a,b) implies (b,a). It is antisymmetric if having both (a,b) and (b,a) forces a=b. Therefore, for two different elements, such as 1 and 2, we cannot include (1,2), because symmetry would require (2,1), while antisymmetry would forbid having both. Thus every off-diagonal pair is impossible. The only pairs that may be included are diagonal pairs: (1,1), (2,2), (3,3), and (4,4). These pairs do not conflict with either property, and including all four gives the maximum. Hence A, C, and D are incorrect: 2 and 8 are not maximum values, while 16 would require forbidden off-diagonal pairs. Memory cue: symmetric allows a pair to be reversed; antisymmetric allows reversal only when the two elements are actually the same.

Correct answer: A. Reflexive

Step 1: For reflexive relation, every element must be related to itself. Step 2: ((1,1),(2,2),(3,3)) are present, so the relation is reflexive. Step 3: In exams, check diagonal pairs first, then test other properties.

Correct answer: C. ({(a,a),(c,c),(a,c)})

Step 1: A reflexive relation must contain ((a,a),(b,b),(c,c)). Step 2: The third option misses ((b,b)), so it is not reflexive. Step 3: If even one self-pair is missing, the relation is not reflexive.

Correct answer: B. (4)

Step 1: The smallest reflexive relation contains only self-pairs. Step 2: For four elements, the pairs are ((1,1),(2,2),(3,3),(4,4)). Step 3: The number of pairs in the smallest reflexive relation equals the number of elements.

Correct answer: D. (25)

Step 1: The universal relation is (A\times A). Step 2: If (A) has (5) elements, then (A\times A) has (5^2=25) pairs. Step 3: For a universal relation, remember the count is (n^2).

Correct answer: B. It is symmetric

Step 1: The empty relation has no ordered pair. Step 2: There is no pair that violates symmetry, so it is considered symmetric. Step 3: Do not confuse it with reflexive, because self-pairs are absent.

Correct answer: C. Identity relation

Step 1: An identity relation contains only self-pairs of the form ((a,a)). Step 2: Here both (1) and (2) have their self-pairs and no extra pair is present. Step 3: Identity relation is also reflexive, but its specific name is identity relation.

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

Step 1: In a symmetric relation, if ((a,b)) is present, then ((b,a)) must also be present. Step 2: The reverse of ((2,3)) is ((3,2)). Step 3: For symmetry, focus on reverse pairs, not only self-pairs.

Correct answer: B. Because ((3,2)) is missing

Step 1: In a symmetric relation, every pair must have its reverse pair. Step 2: ((2,3)) is present but ((3,2)) is missing, so symmetry fails. Step 3: One missing reverse pair is enough to make the relation not symmetric.

Correct answer: A. ((5,4)\in R)

Step 1: Symmetry directly means the reverse ordered pair must exist. Step 2: Since ((4,5)) is present, ((5,4)) must be present. Step 3: Symmetry does not guarantee self-pairs, so read options carefully.

Correct answer: C. Example of transitivity

Step 1: Transitivity means if ((a,b)) and ((b,c)) are present, then ((a,c)) must be present. Step 2: Here ((1,2)) and ((2,3)) are present along with ((1,3)). Step 3: To test transitivity, combine linked pairs and look for the third pair.

Correct answer: B. It is reflexive, symmetric and transitive

Step 1: All self-pairs are present, so the relation is reflexive. Step 2: The reverse of each self-pair is itself, and transitivity is also satisfied. Step 3: An identity relation is usually reflexive, symmetric and transitive.

Correct answer: B. Equivalence relation

Step 1: An equivalence relation has three conditions: reflexive, symmetric and transitive. Step 2: The question gives exactly these three properties. Step 3: For exams, remember these three words together for equivalence relation.

Correct answer: B. It is reflexive, symmetric and transitive

Step 1: (A\times A) contains all possible ordered pairs. Step 2: So all self-pairs, reverse pairs and required transitive pairs are present. Step 3: The universal relation can also be an equivalence relation.

Correct answer: B. Identity relation

Step 1: The condition (a=b) gives only self-pairs. Step 2: So the relation is limited to ((1,1),(2,2),(3,3)). Step 3: Whenever you see (a=b), identify the identity relation.

Correct answer: A. It is reflexive

Step 1: Every natural number divides itself. Step 2: Hence (aRa) is true for every (a), so the relation is reflexive. Step 3: Divisibility is reflexive, but it is generally not symmetric.

Correct answer: C. Equivalence relation

Step 1: (a-a=0) is even, so reflexivity holds. Step 2: If (a-b) is even, then (b-a) is also even, so symmetry holds. Step 3: Adding two even differences gives an even difference, so it is an equivalence relation.

Correct answer: A. Because (a-a=0) is not odd

Step 1: For reflexivity, (aRa) must be true for every (a). Step 2: Here (a-a=0), and (0) is not odd. Step 3: To test self-pairs, substitute (a-a).

Correct answer: A. It is reflexive and transitive

Step 1: (a\le a) is always true, so it is reflexive. Step 2: If (a\le b) and (b\le c), then (a\le c), so it is transitive. Step 3: Do not treat (\le) as symmetric, because (2\le3) but (3\le2) is false.

Correct answer: C. It is transitive

Step 1: (a<a) is never true, so it is not reflexive. Step 2: (a<b) does not imply (b<a), so it is not symmetric. Step 3: If (a<b) and (b<c), then (a<c), so it is transitive.

Correct answer: A. Reflexive but not symmetric

Step 1: All self-pairs are present, so the relation is reflexive. Step 2: ((1,2)) is present but ((2,1)) is missing, so it is not symmetric. Step 3: Test reflexivity and symmetry separately.

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

Step 1: For symmetry, reverse pairs must be present together. Step 2: ((1,2)) and ((2,1)) are both present, but not all self-pairs are present. Step 3: A symmetric relation need not be reflexive.

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

Step 1: Reflexivity requires the self-pair of every element of (A). Step 2: (3) is in the set, but ((3,3)) is not in the relation. Step 3: List all elements and tick their self-pairs.

Correct answer: A. Always reflexive

Step 1: Both reflexive relations contain every ((a,a)). Step 2: Therefore (R\cap S) also contains every ((a,a)). Step 3: Intersection keeps pairs common to both, so self-pairs remain.