Reflexive relation

Correct answer: B. Only ((3,3))

Step 1: A relation is reflexive when every (a \in A) has ((a,a)) in the relation. Step 2: ((1,1)) and ((2,2)) are present but ((3,3)) is missing. Step 3: In exams, check diagonal pairs first.

Correct answer: C. (2^{12}=4096)

Step 1: With (4) elements, there are (4^2=16) ordered pairs. Step 2: The (4) diagonal pairs are compulsory, so (16-4=12) pairs are optional. Step 3: Use (2^{n^2-n}) for quick counting.

Correct answer: C. (^{20}C_3)

Step 1: There are (5^2=25) total pairs and (5) diagonal pairs are compulsory. Step 2: There are (25-5=20) non-diagonal pairs, and exactly (3) must be chosen. Step 3: Separate compulsory and optional pairs before counting.

Correct answer: C. Contains all ((p,p),(q,q),(r,r),(s,s))

Step 1: In a reflexive relation, each element of the set must be related to itself. Step 2: For (A), all four diagonal pairs are required. Step 3: Extra pairs do not matter, but all diagonal pairs must be present.

Correct answer: B. (1)

Step 1: To check reflexivity from a matrix, look at the main diagonal. Step 2: The diagonal entries are (1,0,1), so the middle entry must be changed to (1). Step 3: Only the missing diagonal entry needs correction.

Correct answer: A. (R \cap S) is always reflexive

Step 1: Every ((a,a)) is present in both (R) and (S). Step 2: A pair common to both remains in (R \cap S), so all diagonal pairs remain. Step 3: The intersection of reflexive relations is reflexive.

Correct answer: A. (R^{-1}) is always reflexive

Step 1: Since (R) is reflexive, every ((a,a) \in R). Step 2: The inverse of ((a,a)) is again ((a,a)). Step 3: Diagonal pairs remain unchanged in the inverse relation.

Correct answer: B. (R \cup \Delta_A)

Step 1: (\Delta_A={(a,a):a \in A}) contains all diagonal pairs. Step 2: (R \cup \Delta_A) keeps the old pairs and adds any missing diagonal pairs. Step 3: This gives the smallest reflexive closure.

Correct answer: C. (R) is reflexive because every (a+a) is even

Step 1: To test reflexivity, check ((a,a)). Step 2: For every (a), (a+a=2a) is even, so ((a,a) \in R). Step 3: In such questions, put (b=a) first.

Correct answer: B. Yes, because (a-a=0) and (0) is divisible by (4)

Step 1: For reflexivity, every ((a,a)) must belong to the relation. Step 2: (a-a=0), and (0) is divisible by any non-zero integer. Step 3: In divisibility-type relations, the diagonal difference is always (0).

Correct answer: C. (R) is not reflexive because no ((a,a)) belongs to it

Step 1: Put (a=b) for diagonal pairs. Step 2: Then (a+b=2a), which is always even, not odd. Step 3: If required diagonal pairs are missing, the relation is not reflexive.

Correct answer: A. Because every (a \le a) is true

Step 1: Reflexivity checks whether each element is related to itself. Step 2: For every number, (a \le a) is true, so ((a,a)) is included. Step 3: Be careful about the difference between (<) and (\le).

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

Step 1: Reflexivity requires ((a,a)). Step 2: The condition (a<a) is false for every (a), so no diagonal pair appears. Step 3: Strict inequality relations are usually not reflexive.

Correct answer: A. Reflexive because (X \subseteq X)

Step 1: Each element of (P(A)) is itself a set. Step 2: For every set (X), (X \subseteq X) is true. Step 3: Remember the difference between subset and proper subset.

Correct answer: B. It is not reflexive because (X \subset X) is false

Step 1: A proper subset does not allow equality. Step 2: No set is a proper subset of itself. Step 3: Distinguish (\subseteq) from (\subset) carefully.

Correct answer: A. Reflexive because (a) divides itself

Step 1: For reflexivity, check each ((a,a)). Step 2: Every non-zero number divides itself, so ((a,a)) is included. Step 3: Missing non-diagonal pairs do not affect reflexivity.

Correct answer: C. No, because (\gcd(2,2)=2) and (\gcd(3,3)=3)

Step 1: Reflexivity requires ((1,1),(2,2),(3,3)). Step 2: (\gcd(1,1)=1), but (\gcd(2,2)=2) and (\gcd(3,3)=3). Step 3: If even one diagonal pair is missing, the relation is not reflexive.

Correct answer: A. Because (a^2=a^2) for every (a)

Step 1: On the diagonal, put (b=a). Step 2: The condition becomes (a^2=a^2), which is always true. Step 3: Reflexivity only needs each element to be related to itself.

Correct answer: C. No, because ((-1,-1)) and ((1,1)) are absent

Step 1: Reflexivity requires every diagonal pair. Step 2: ((0,0)) satisfies the condition, but ((-1,-1)) and ((1,1)) do not. Step 3: Having only some diagonal pairs is not enough.

Correct answer: A. Reflexive because (|a-a|=0\le 2)

Step 1: Check the condition for each ((a,a)). Step 2: (|a-a|=0), and (0\le 2) is true. Step 3: Far non-diagonal pairs do not affect reflexivity.

Correct answer: B. Not reflexive because (|a-a|=0)

Step 1: In a diagonal pair, both entries are the same. Step 2: Therefore (|a-a|=0), which is not greater than (0). Step 3: Conditions like (>0) often remove self-related pairs.

Correct answer: A. (R) is reflexive because there is no element to check

Step 1: Reflexivity says every element must be related to itself. Step 2: The empty set has no element, so the condition is not violated. Step 3: Remember this as a vacuous truth.

Correct answer: B. Not reflexive because no ((a,a)) is present

Step 1: A non-empty (A) has at least one element (a). Step 2: Reflexivity requires ((a,a)), but the empty relation has no pairs. Step 3: Distinguish an empty set from an empty relation on a non-empty set.

Correct answer: A. It is always reflexive

Step 1: (A \times A) contains all possible ordered pairs from (A). Step 2: Hence every ((a,a)) is definitely included. Step 3: The universal relation automatically satisfies reflexivity.

Correct answer: C. It is not reflexive because no diagonal pair remains

Step 1: Since (R) is reflexive, all ((a,a)) are in (R). Step 2: In the complement (A \times A-R), those diagonal pairs are removed. Step 3: On a non-empty set, missing diagonal pairs destroy reflexivity.