Reflexive relation

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

Step 1: The required self-pairs are ((2,2),(3,3),(4,4)). Step 2: ((3,3)) is not present in the given (R). Step 3: In a reflexive relation, every element must be connected to itself.

Correct answer: A. ((a,a) \in R)

Step 1: The rule of reflexivity applies to every element. Step 2: If (a \in A), then ((a,a)) must belong to (R). Step 3: In symbol-based questions, focus on self-pairs.

Correct answer: A. Yes, because (R) is the universal relation

Step 1: (A \times A) contains all ordered pairs from (A). Step 2: Since (2 \in A), ((2,2)) belongs to (A \times A). Step 3: The universal relation is a simple example of a reflexive relation.

Correct answer: A. Reflexive

Step 1: When (a=b), the ordered pair becomes a self-pair. Step 2: For every (a \in A), ((a,a)) satisfies this rule. Step 3: The relation (a=b) is the identity relation and is reflexive.

Correct answer: A. Yes, because (a \le a) is true for every (a)

Step 1: For reflexivity, check the case (a=a). Step 2: Every number is equal to itself, so (a \le a) is true. Step 3: In inequality relations, first test the equality case.

Correct answer: A. Because (a<a) is never true

Step 1: Reflexivity requires ((a,a)). Step 2: The rule (a<b) would require (a<a), which is false. Step 3: Strict inequality relations are usually not reflexive.

Correct answer: A. Yes, because (a-a=0) is even

Step 1: For reflexivity, put (a=b). Step 2: Then (a-a=0), and 0 is even. Step 3: In difference-based rules, first subtract the element from itself.

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

Step 1: For a self-pair, take (a=b). Step 2: Then (a-a=0), and 0 is not odd. Step 3: Reflexivity needs self-pairs for all elements.

Correct answer: A. Yes, because every (a) divides itself

Step 1: For reflexivity, check whether (a) divides itself. Step 2: Since (a=a \times 1), (a) divides itself. Step 3: In divisibility, remember the self-divisibility case.

Correct answer: A. Because (a\ne a) is false

Step 1: Reflexivity requires ((a,a)). Step 2: But under (a\ne b), (a\ne a) is false, so self-pairs are absent. Step 3: A not-equal relation is not reflexive.

Correct answer: A. Yes, because (a+a=2a) is even

Step 1: For reflexivity, put ((a,a)) into the rule. Step 2: (a+a=2a) is always even, so every self-pair is included. Step 3: In sum-based rules, quickly test (a+a).

Correct answer: A. No, because (a+a=2a) is even

Step 1: For a self-pair, put (b=a) in (a+b). Step 2: (a+a=2a) is even, not odd. Step 3: So this relation is not reflexive.

Correct answer: A. Reflexive

Step 1: For self-pairs, put (a=b). Step 2: Then (a-a=0), so every ((a,a)) satisfies the rule. Step 3: (a-b=0) represents equality.

Correct answer: A. Yes, because every number has the same remainder as itself

Step 1: For reflexivity, compare each number with itself. Step 2: Every number has the same remainder as itself. Step 3: In same-remainder relations, self-pairs always satisfy the rule.

Correct answer: A. (R) is reflexive

Step 1: For reflexivity, put (a=b). Step 2: For every (a), (a \equiv a \pmod{2}) is true. Step 3: Congruence relations include all self-pairs.

Correct answer: A. No, because ((1,1)) does not satisfy the rule

Step 1: Reflexivity needs ((1,1),(2,2),(3,3),(4,4)). Step 2: The rule requires the first element to be even, but (1) is not even, so ((1,1)) is missing. Step 3: Missing one required pair makes it not reflexive.

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

Step 1: Every reflexive relation contains all self-pairs. Step 2: Since (2 \in A), ((2,2)) must be present in every reflexive relation. Step 3: Pairs with different elements are not always required.

Correct answer: A. ((1,2) \in R)

Step 1: Reflexivity makes only self-pairs compulsory. Step 2: ((1,2)) is not a self-pair, so it is not necessary. Step 3: Pay attention to the phrase not necessary in the question.

Correct answer: A. 4

Step 1: The identity relation contains only self-pairs. Step 2: (A) has 4 elements, so it has 4 pairs. Step 3: The identity relation and minimum reflexive relation have the same count.

Correct answer: A. (R) is the minimum reflexive relation

Step 1: The given (R) has all self-pairs. Step 2: It has no extra pair, so it is the minimum reflexive relation. Step 3: The minimum reflexive relation is also the identity relation.

Correct answer: A. Yes

Step 1: Identify the required self-pairs separately. Step 2: ((1,1),(2,2),(3,3)) are present, so (R) is reflexive. Step 3: Do not change the answer because of extra cyclic pairs.

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

Step 1: All self-pairs of (A) are required. Step 2: For (3), ((3,3)) is necessary but not given. Step 3: Even if other pairs are present, one missing self-pair fails reflexivity.

Correct answer: A. Self-pair of every element

Step 1: Reflexive relation is identified by self-pairs. Step 2: So the first task is to check ((a,a)) for every element. Step 3: Check symmetric or other properties later.

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

Step 1: In a reflexive relation, each element of (A) is related to itself. Step 2: Since (4 \in A), ((4,4)) must be present. Step 3: Do not treat pairs with different elements as compulsory.

Correct answer: A. For some (a \in A), ((a,a) \notin R)

Step 1: Not reflexive means at least one required self-pair is missing. Step 2: So for some (a \in A), ((a,a) \notin R). Step 3: Understanding the opposite condition helps identify the property quickly.