Reflexive relation

Step 1: Reflexivity requires all self-pairs, including ((1,1)). Step 2: The rule requires the first element to be even, but (1) is not even. Step 3: Missing even one self-pair makes the relation non-reflexive.

Step 1: Reflexivity requires every (a) to be related to itself. Step 2: Every natural number (a) divides itself because (a=a\times1). Step 3: In divisibility questions, remember self-divisibility.

Step 1: A proper divisor requires (a\ne b). Step 2: In a self-pair ((a,a)), (a=b), so self-pairs are not included. Step 3: Distinguish proper divisor from ordinary divisibility.

Step 1: Here the elements are subsets, not ordinary numbers. Step 2: Every subset (X) is a subset of itself, so (XRX) is true. Step 3: In subset relations, read the base family carefully.

Step 1: (X\subset Y) means proper subset. Step 2: No set is a proper subset of itself. Step 3: In exams, clearly distinguish (\subset) and (\subseteq).

Step 1: For reflexivity, put (b=a). Step 2: (|a-a|=0) and (0\le0) is true. Step 3: In absolute value relations, the distance from an element to itself is zero.

Step 1: For reflexivity, ((a,a)) must satisfy the rule. Step 2: (|a-a|=0), and (0>0) is false. Step 3: Self-distance is zero, so it fails a strict positive condition.

Step 1: For a self-pair, the sum becomes (a+a). Step 2: (a+a=2a) is always even, whether (a) is even or odd. Step 3: In parity-of-sum relations, first check (2a).

Step 1: For ((a,a)), (a+a=2a) is even, not odd. Step 2: So none of the self-pairs are in the original relation. Step 3: To make it reflexive, add all ((1,1),(2,2),(3,3),(4,4)).

Step 1: Both relations contain ((1,1),(2,2),(3,3)). Step 2: The intersection keeps pairs common to both, so all self-pairs remain. Step 3: The intersection of two reflexive relations is reflexive.

Step 1: Both (R) and (S) contain all self-pairs of (A). Step 2: The union contains all pairs from both relations, so the self-pairs remain. Step 3: The union of reflexive relations is always reflexive.

Step 1: (S) contains all self-pairs. Step 2: In (R-S), pairs of (S) are removed, so self-pairs may also be removed. Step 3: In difference of relations, check whether compulsory pairs are being removed.

Step 1: Since (R) is reflexive, (R) contains all self-pairs. Step 2: Because (R\subseteq T), those self-pairs are also in (T). Step 3: A super-relation of a reflexive relation within (A\times A) is also reflexive.

Step 1: (R) has all self-pairs, but (T) may be a smaller part of (R). Step 2: Some self-pairs may be removed while forming (T). Step 3: Reflexivity is not automatically preserved in sub-relations.

Step 1: (R) already contains all self-pairs. Step 2: Adding an extra pair does not remove these required pairs. Step 3: Reflexivity is not destroyed by adding pairs within (A\times A).

Step 1: The identity relation contains all self-pairs. Step 2: Removing ((3,3)) means (3) is not related to itself. Step 3: Removing even one compulsory pair destroys reflexivity.

Step 1: (A\times A) contains all self-pairs. Step 2: Removing ((2,2)) removes one required pair. Step 3: If a self-pair is removed from the universal relation, reflexivity fails.

Step 1: The removed pairs ((1,2)) and ((2,3)) are not self-pairs. Step 2: ((1,1),(2,2),(3,3)) still remain in the relation. Step 3: Removing only non-diagonal pairs does not break reflexivity.

Step 1: The minimum reflexive relation has 4 pairs. Step 2: The universal relation (A\times A) has (4^2=16) pairs. Step 3: The difference is (16-4=12).

Step 1: The number of all relations is (2^{25}). Step 2: The number of reflexive relations is (2^{25-5}=2^{20}). Step 3: The ratio is (2^{25}:2^{20}=2^5:1).