Reflexive relation

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

Answer: A. A relation R on A is reflexive when every element is related to itself. Therefore, for A={1,2,3}, the compulsory diagonal ordered pairs are (1,1), (2,2), and (3,3). A relation may also contain extra pairs, so the additional pair (1,2) in option A does not cause any problem. Option B is not reflexive because (3,3) is missing. Option C has none of the three required diagonal pairs, so it fails immediately. Option D contains (1,1) and (3,3), but it omits (2,2), so it also fails. A useful memory cue is: reflexive means every element points back to itself; first check the diagonal pairs, and only then consider extra pairs.

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

Step 1: In a reflexive relation, every element must be related to itself. Step 2: So ((a,a)) must be in (R) for every (a \in A). Step 3: In exams, first check all diagonal pairs.

Correct answer: A. ((1,1),(2,2),(3,3))

Step 1: A reflexive relation needs each element paired with itself. Step 2: For (1,2,3), the required pairs are ((1,1),(2,2),(3,3)). Step 3: Extra pairs may be present, but these are compulsory.

Correct answer: A. Reflexive

Step 1: Check self-pairs for all elements of (A). Step 2: ((1,1),(2,2),(3,3)) are all in (R), so the relation is reflexive. Step 3: The extra pair ((1,2)) does not stop reflexivity.

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

Step 1: A reflexive relation needs every self-pair. Step 2: Since (3 \in A), ((3,3)) is required, but it is not in (R). Step 3: If even one required self-pair is missing, the relation is not reflexive.

Correct answer: A. 4

Step 1: A reflexive relation must contain one self-pair for each element. Step 2: For 4 elements, at least 4 diagonal pairs are needed. Step 3: For minimum count, count only compulsory pairs.

Correct answer: A. (R={(a,a),(b,b),(c,c),(d,d)})

Step 1: The minimum reflexive relation contains only the required self-pairs. Step 2: For (a,b,c,d), the pairs ((a,a),(b,b),(c,c),(d,d)) are needed. Step 3: The key check is the presence of all diagonal pairs.

Correct answer: A. Reflexive

Step 1: (A \times A) contains all possible ordered pairs from (A). Step 2: Therefore every ((a,a)) is included. Step 3: When judging universal relation, check the base set.

Correct answer: A. Because required ((a,a)) pairs are absent

Step 1: A non-empty set (A) has at least one element. Step 2: The empty relation has no pairs, so the self-pair of that element is missing. Step 3: Do not mark the empty relation reflexive on a non-empty set.

Correct answer: A. (R) is reflexive

Step 1: (A) has only the element (5). Step 2: Reflexivity needs ((5,5)), and it is present in (R). Step 3: A one-element set has only one compulsory self-pair.

Correct answer: A. Not reflexive

Step 1: Since (5) is in (A), ((5,5)) is required. Step 2: (R) is empty, so ((5,5)) is not present. Step 3: Reflexivity fails even if one required pair is missing.

Correct answer: A. 2

Step 1: A reflexive relation needs the self-pair of each element. Step 2: For (1) and (2), ((1,1)) and ((2,2)) are required. Step 3: The minimum number equals the number of elements.

Correct answer: A. (R={(1,1),(1,2)})

Step 1: For (A={1,2}), both ((1,1)) and ((2,2)) are needed. Step 2: The first relation misses ((2,2)). Step 3: Check missing compulsory pairs before looking at extra pairs.

Correct answer: A. Identity relation and reflexive relation

Step 1: (R) contains each element paired with itself. Step 2: Such a relation is the identity relation and it is reflexive. Step 3: The identity relation on a set is always reflexive.

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

Step 1: In a reflexive relation, every element is related to itself. Step 2: Since (4 \in A), ((4,4)) is compulsory. Step 3: Pairs with different elements are not compulsory for reflexivity.

Correct answer: A. Both ((p,p)) and ((q,q)) will be in (R)

Step 1: Reflexivity depends on self-pairs. Step 2: For (p) and (q), ((p,p)) and ((q,q)) are required. Step 3: The same rule works for sets with letters.

Correct answer: A. It is reflexive

Step 1: Reflexivity only requires all self-pairs to be present. Step 2: ((1,1),(2,2),(3,3)) are present, so (R) is reflexive. Step 3: Do not get confused by extra pairs.

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

Step 1: The required pairs for (A) are ((1,1),(2,2),(3,3)). Step 2: ((2,2)) is missing from the given (R). Step 3: Add the missing diagonal pair to make it reflexive.

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

Step 1: The elements (x,y,z) need their self-pairs. Step 2: ((x,x)) and ((y,y)) are present, but ((z,z)) is missing. Step 3: Add the missing self-pair of the missing element.

Correct answer: A. (A \times A)

Step 1: A relation on (A) is always a subset of (A \times A). Step 2: A reflexive relation is also a relation, so it is a subset of (A \times A). Step 3: First recall the basic definition of a relation.

Correct answer: A. 5

Step 1: The smallest reflexive relation contains only diagonal pairs. Step 2: (A) has 5 elements, so 5 self-pairs are required. Step 3: For minimum reflexive relation, the number is (n).

Correct answer: A. (n)

Step 1: Each element needs one self-pair. Step 2: For (n) elements, there are (n) such pairs. Step 3: (n^2) is the number of all possible pairs, not the minimum reflexive relation.

Correct answer: A. (n^2)

Step 1: In (A \times A), each element can appear in the first and second position. Step 2: So total pairs are (n \times n=n^2). Step 3: Reflexive relation needs (n) compulsory pairs, while (A \times A) has (n^2) possible pairs.

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

Step 1: Reflexivity requires only self-pairs. Step 2: ((1,2)) has two different elements, so it is not compulsory. Step 3: Separate compulsory pairs from optional pairs.

Correct answer: A. Reflexive

Step 1: Look for the self-pairs of (2,3,4). Step 2: ((2,2),(3,3),(4,4)) are all present, so (R) is reflexive. Step 3: Extra reverse pairs do not affect the reflexive check.