Reflexive relation

Correct answer: C. (4)

Step 1: (A\times A) has (4) pairs. Step 2: ((1,1)) and ((2,2)) are compulsory, while the remaining (2) pairs are optional. Step 3: Hence the number is (2^2=4).

Correct answer: B. (2^6)

Step 1: Total pairs are (3^2=9). Step 2: The (3) self-pairs are compulsory, so (9-3=6) pairs are optional. Step 3: Each optional pair has two choices, so (2^6) relations are possible.

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

Step 1: In a reflexive relation, each element is related to itself. Step 2: Since (3) is an element of the set, ((3,3)) is compulsory. Step 3: Pairs with other elements are not compulsory.

Correct answer: A. Yes

Step 1: Reflexivity only requires all self-pairs. Step 2: All four required self-pairs are present. Step 3: Whether extra pairs exist or not, the relation is reflexive.

Correct answer: A. Every element is related to itself

Step 1: Reflexive means returning to the same element. Step 2: Therefore every element must be related to itself. Step 3: For definition-based questions, remember this key sentence.

Correct answer: A. The relation remains reflexive

Step 1: All three self-pairs needed for reflexivity are present. Step 2: ((2,1)) is an extra pair and does not remove reflexivity. Step 3: Do not worry about extra pairs; check the required pairs.

Correct answer: A. At least one ((a,a)) pair is missing

Step 1: Reflexivity fails when a required self-pair is absent. Step 2: If ((a,a)\notin R) for some (a\in A), then (R) is not reflexive. Step 3: To find the failure, check each diagonal pair.

Correct answer: A. Because all self-pairs are present

Step 1: Each element of (A) needs its self-pair. Step 2: ((1,1),(2,2),(3,3)) are all in (R). Step 3: Reflexivity is decided by these pairs.

Correct answer: A. It is reflexive

Step 1: In the identity relation, each element is related to itself. Step 2: So ((a,a)) is present for every (a\in A). Step 3: The identity relation is a basic example of a reflexive relation.

Correct answer: B. (R) is not reflexive

Step 1: The set (A) has three elements (1,2,3). Step 2: ((3,3)) is also required, but it is absent. Step 3: If one element misses its self-pair, the relation is not reflexive.

Correct answer: A. Reflexive

Step 1: For ((a,a)), we get (a-a=0). Step 2: (0) is even, so every self-pair satisfies the condition. Step 3: For subtraction conditions, test by putting (a=a).

Correct answer: B. (R) is not reflexive

Step 1: For a self-pair, (a-b=a-a=0). Step 2: (0) is not odd, so ((a,a)) does not satisfy the condition. Step 3: Therefore the relation is not reflexive.

Correct answer: A. Yes

Step 1: For reflexivity, check (a\ge a). Step 2: Every number is equal to itself, so (a\ge a) is true. Step 3: The relation (\ge) is reflexive.

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

Step 1: For a self-pair, put (b=a) in (a>b). Step 2: This gives (a>a), which is false. Step 3: Hence the relation (>) is not reflexive.

Correct answer: A. Reflexive

Step 1: The four elements of (A) need their self-pairs. Step 2: ((1,1),(2,2),(3,3),(4,4)) are all given. Step 3: ((1,4)) is extra and does not affect reflexivity.

Correct answer: A. (n)

Step 1: The smallest reflexive relation contains only self-pairs. Step 2: For (n) elements, there are (n) such pairs. Step 3: The minimum number of pairs is always (n).

Correct answer: B. (n^2)

Step 1: The universal relation is (A\times A). Step 2: With (n) elements, it has (n^2) ordered pairs. Step 3: It is also reflexive because it contains all self-pairs.

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

Step 1: For (A={1,2}), ((1,1)) and ((2,2)) are required. Step 2: The third relation does not contain ((2,2)). Step 3: Therefore it is not reflexive.

Correct answer: A. Yes

Step 1: The condition (a=b) gives every self-pair. Step 2: ((1,1),(2,2),(3,3)) will all be in (R). Step 3: The other condition may add pairs, but reflexivity remains.

Correct answer: B. It is not reflexive

Step 1: In a self-pair, (a=b). Step 2: The condition (a\ne b) excludes all self-pairs. Step 3: Therefore the relation is not reflexive.

Correct answer: A. ((4,4),(5,5),(6,6)\in R)

Step 1: If a relation is reflexive, every element is related to itself. Step 2: For (4,5,6), the pairs ((4,4),(5,5),(6,6)) must be present. Step 3: Reflexivity alone does not decide other pairs.

Correct answer: D. Any of these

Step 1: (R) is already reflexive because all self-pairs are present. Step 2: Adding any extra pair does not remove those pairs. Step 3: Reflexivity is not spoiled by extra pairs.

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

Step 1: For (1,2,3), the pairs ((1,1),(2,2),(3,3)) are needed. Step 2: The first two are present, but ((3,3)) is missing. Step 3: Add this pair to make the relation reflexive.

Correct answer: D. (4)

Step 1: There is one self-pair for each element. Step 2: (A) has (4) elements, so (4) such pairs are required. Step 3: When counting, directly look at the number of elements.

Correct answer: A. Yes, because reflexivity only needs all ((a,a)) pairs

Step 1: Reflexivity does not require all possible pairs. Step 2: It only requires the self-pair of every element. Step 3: So the relation can be reflexive even if some other pairs are missing.