Reflexive relation

Correct answer: A. Both are reflexive

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.

Correct answer: A. Minimum (4), maximum (16)

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.

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

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.

Correct answer: B. (4)

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.

Correct answer: B. (R={(1,1),(2,2),(3,3),(1,2)})

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.

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

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.

Correct answer: A. Reflexive

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.

Correct answer: A. Because (a+a=2a) is always even

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.

Correct answer: B. (R) is not reflexive

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.

Correct answer: A. (2^{20})

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}).

Correct answer: C. (n^2-n)

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.

Correct answer: B. No, because the last main diagonal entry is (0)

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.

Correct answer: A. The relation is reflexive

Step 1: Look at the main diagonal entries. Step 2: Here the main diagonal is (1,1,1), so all ((a,a)) pairs are present. Step 3: If the main diagonal is complete, extra pairs do not spoil reflexivity.

Correct answer: A. Yes

Step 1: For reflexivity, check (|a-a|). Step 2: (|a-a|=0), and (0\le 1) is true. Step 3: In absolute value relations, test self-pairs separately.

Correct answer: B. (R) is not reflexive

Step 1: For reflexivity, check (|a-a|). Step 2: (|a-a|=0), but the condition requires (1). Step 3: The self-pairs do not satisfy the condition, so the relation is not reflexive.

Correct answer: A. Yes

Step 1: Every number has the same parity as itself. Step 2: Therefore, for every (a\in A), ((a,a)) belongs to the relation. Step 3: Relations based on having the same property often become reflexive.

Correct answer: A. Because every (a) divides itself

Step 1: For reflexivity, check (a\mid a). Step 2: Every positive number divides itself. Step 3: In divisibility relations, first test the self-case.

Correct answer: B. No

Step 1: For reflexivity, ((3,3)) must also be present. Step 2: (3+3=6), which is not less than or equal to (4). Step 3: If even one self-pair fails, the relation is not reflexive.

Correct answer: A. (R) is reflexive

Step 1: For self-pairs, (a+a=2a). Step 2: For (a=1,2,3), (2a\ge 2) is true. Step 3: If all self-pairs satisfy the condition, the relation is reflexive.

Correct answer: C. (4)

Step 1: A reflexive relation needs one self-pair for each element. Step 2: (A) has four elements, so four self-pairs are compulsory. Step 3: Total pairs may be more, but compulsory self-pairs equal the number of elements.

Correct answer: B. Every identity relation is reflexive

The governing concept is reflexivity. A relation R on a set A is reflexive when (a,a) belongs to R for every element a in A. The identity relation I_A is defined as I_A = {(a,a) : a is in A}; therefore it contains every required self-pair and is always reflexive. However, a reflexive relation may contain additional pairs such as (a,b), so it need not be the identity relation. On a non-empty set, the empty relation cannot be reflexive because it contains none of the required self-pairs. Thus option B is correct, while A reverses the implication and C and D contradict the definition.

Correct answer: A. (R) is reflexive and not the identity relation

Step 1: ((1,1),(2,2),(3,3)) are all present, so the relation is reflexive. Step 2: ((1,2)) and ((2,1)) are extra pairs. Step 3: With extra pairs, the relation is not identity, but it can still be reflexive.

Correct answer: B. (R) is not reflexive because ((2,2)) is missing

Step 1: A reflexive relation needs self-pairs for (1,2,3). Step 2: Here ((2,2)) is missing. Step 3: Do not get confused by extra pairs; find the missing diagonal pair.

Correct answer: A. Reflexive

Step 1: The condition includes (a=b). Step 2: Therefore, for every (a\in A), ((a,a)) belongs to the relation. Step 3: If equality is part of the condition, reflexivity is likely and should be checked first.

Correct answer: B. No

Step 1: For reflexivity, the condition would require (a=a+1). Step 2: No number can be one more than itself. Step 3: If self-pairs fail the condition, the relation is not reflexive.