Reflexive relation

Step 1: In a self-pair, the sum is (2a). Step 2: For (a=1), (2a=2), which is not divisible by (3). Step 3: One failed element is enough to disprove reflexivity.

Step 1: The self-pairs are ((1,1),(2,2),(3,3)). Step 2: Each has sum (2a), which is divisible by (2). Step 3: Therefore all three self-pairs belong to (R).

Step 1: The condition (a=b) includes every self-pair. Step 2: Thus ((1,1),(2,2),(3,3)) are all in (R). Step 3: The second condition may add pairs, but reflexivity remains safe.

Step 1: For ((1,1)), (a\ne b) is false and (a=2) is also false. Step 2: So ((1,1)) does not belong to (R). Step 3: Missing one self-pair makes the relation not reflexive.

Step 1: For a self-pair, put (b=a). Step 2: The condition becomes (a\le a+1), which is true for every (a). Step 3: In such questions, substituting (b=a) is the direct method.

Step 1: For a self-pair, put (b=a). Step 2: The condition becomes (a+1\le a), which is false for every number. Step 3: So no self-pair is included and (R) is not reflexive.

Step 1: All elements of (A) are positive. Step 2: For a self-pair, (ab=a^2), which is greater than (0) for every (a\in A). Step 3: In product conditions, check (a^2) for reflexivity.

Step 1: Reflexivity requires the self-pair of every element. Step 2: For (0), the pair ((0,0)) gives product (0), which is not greater than (0). Step 3: Be careful with product conditions when zero is in the set.

Step 1: In (R), putting ((a,a)) gives (a+a=2a), which is even. Step 2: In (S), putting ((a,a)) gives (a-a=0), which is even. Step 3: Both relations contain all self-pairs.

Step 1: Reflexivity requires four self-pairs, so the minimum is (4). Step 2: The maximum relation is (A\times A), which has (4^2=16) pairs. Step 3: Count compulsory pairs and total pairs separately.

Step 1: A reflexive relation must contain ((a,a)) for every element of (A). Step 2: The pairs for (1) and (2) are present, but ((3,3)) is missing. Step 3: In exams, first find the missing self-pair.

Step 1: In a reflexive relation, every element is related to itself. Step 2: The set (A) has four elements, so ((a,a),(b,b),(c,c),(d,d)) are compulsory. Step 3: When the minimum is asked, count only compulsory pairs.

Step 1: Reflexivity requires all three self-pairs. Step 2: The second option has all of them and also one extra pair, so it is not only the identity relation. Step 3: Understand the difference between identity and reflexive relations through extra pairs.

Step 1: Reflexivity requires every element to be related to itself. Step 2: For any (a), (a\le a) is true, so ((a,a)) belongs to the relation. Step 3: When equality is included in an inequality, reflexivity can be checked quickly.

Step 1: To check reflexivity, test (a-a). Step 2: (a-a=0), and (0) is divisible by (2). Step 3: If the condition is true for every element with itself, the relation is reflexive.

Step 1: For reflexivity, put ((a,a)). Step 2: Then (a+a=2a), which is always even. Step 3: In such questions, check the self-pair, not a general pair first.

Step 1: For reflexivity, check (a+a). Step 2: (a+a=2a) is always even, not odd. Step 3: If ((a,a)) fails the condition, the relation is not reflexive.

Step 1: (A\times A) has (5^2=25) pairs. Step 2: The (5) self-pairs are compulsory, so (20) pairs are optional. Step 3: Total reflexive relations are (2^{20}).

Step 1: (A\times A) contains (n^2) pairs. Step 2: The (n) pairs ((a,a)) are compulsory, so they are not freely chosen. Step 3: The remaining number is (n^2-n), which may be included or excluded.

Step 1: In a matrix, reflexivity is checked from the main diagonal. Step 2: The main diagonal is (1,1,0), so ((3,3)) is missing. Step 3: In matrix questions, inspect the main diagonal before the whole table.