Reflexive relation

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Step 1: In a reflexive relation, only self-pairs are compulsory. Step 2: Since (4\in A), ((4,4)) must be present. Step 3: Other pairs of the form ((a,b)) are not compulsory.

Step 1: (A\times A) has (3^2=9) pairs. Step 2: The (3) diagonal pairs are compulsory, so (6) pairs are optional. Step 3: Total reflexive relations are (2^6=64).

Step 1: Check the self-pairs of (x,y,z). Step 2: ((x,x),(y,y),(z,z)) are all present. Step 3: ((x,y)) is extra and does not affect reflexivity.

Step 1: A reflexive relation needs self-pairs for (x,y,z). Step 2: ((x,x)) and ((z,z)) are present, but ((y,y)) is missing. Step 3: Add the self-pair of the missing element.

Step 1: The second relation is non-empty because it has pairs. Step 2: But it does not contain ((1,1),(2,2),(3,3)), so it is not reflexive. Step 3: Keep the difference between non-empty and reflexive clear.

Step 1: In an arrow diagram, an arrow from a point to itself represents ((a,a)). Step 2: Since every point has such an arrow, every element is related to itself. Step 3: A loop at every point is the sign of reflexivity in a diagram.

Step 1: A reflexive relation needs a self-arrow at every element. Step 2: Element (3) has no such arrow, so ((3,3)) is missing. Step 3: If one point lacks a loop, reflexivity fails.