Reflexive relation

Step 1: In a reflexive relation, every element must be related to itself. Step 2: So ((a,a)) must be in (R) for every (a \in A). Step 3: In exams, first check all diagonal pairs.

Step 1: A reflexive relation needs each element paired with itself. Step 2: For (1,2,3), the required pairs are ((1,1),(2,2),(3,3)). Step 3: Extra pairs may be present, but these are compulsory.

Step 1: Check self-pairs for all elements of (A). Step 2: ((1,1),(2,2),(3,3)) are all in (R), so the relation is reflexive. Step 3: The extra pair ((1,2)) does not stop reflexivity.

Step 1: A reflexive relation needs every self-pair. Step 2: Since (3 \in A), ((3,3)) is required, but it is not in (R). Step 3: If even one required self-pair is missing, the relation is not reflexive.

Step 1: A reflexive relation must contain one self-pair for each element. Step 2: For 4 elements, at least 4 diagonal pairs are needed. Step 3: For minimum count, count only compulsory pairs.

Step 1: The minimum reflexive relation contains only the required self-pairs. Step 2: For (a,b,c,d), the pairs ((a,a),(b,b),(c,c),(d,d)) are needed. Step 3: The key check is the presence of all diagonal pairs.

Step 1: (A \times A) contains all possible ordered pairs from (A). Step 2: Therefore every ((a,a)) is included. Step 3: When judging universal relation, check the base set.

Step 1: A non-empty set (A) has at least one element. Step 2: The empty relation has no pairs, so the self-pair of that element is missing. Step 3: Do not mark the empty relation reflexive on a non-empty set.

Step 1: (A) has only the element (5). Step 2: Reflexivity needs ((5,5)), and it is present in (R). Step 3: A one-element set has only one compulsory self-pair.

Step 1: Since (5) is in (A), ((5,5)) is required. Step 2: (R) is empty, so ((5,5)) is not present. Step 3: Reflexivity fails even if one required pair is missing.

Step 1: A reflexive relation needs the self-pair of each element. Step 2: For (1) and (2), ((1,1)) and ((2,2)) are required. Step 3: The minimum number equals the number of elements.

Step 1: For (A={1,2}), both ((1,1)) and ((2,2)) are needed. Step 2: The first relation misses ((2,2)). Step 3: Check missing compulsory pairs before looking at extra pairs.

Step 1: (R) contains each element paired with itself. Step 2: Such a relation is the identity relation and it is reflexive. Step 3: The identity relation on a set is always reflexive.

Step 1: In a reflexive relation, every element is related to itself. Step 2: Since (4 \in A), ((4,4)) is compulsory. Step 3: Pairs with different elements are not compulsory for reflexivity.

Step 1: Reflexivity depends on self-pairs. Step 2: For (p) and (q), ((p,p)) and ((q,q)) are required. Step 3: The same rule works for sets with letters.

Step 1: Reflexivity only requires all self-pairs to be present. Step 2: ((1,1),(2,2),(3,3)) are present, so (R) is reflexive. Step 3: Do not get confused by extra pairs.

Step 1: The required pairs for (A) are ((1,1),(2,2),(3,3)). Step 2: ((2,2)) is missing from the given (R). Step 3: Add the missing diagonal pair to make it reflexive.

Step 1: The elements (x,y,z) need their self-pairs. Step 2: ((x,x)) and ((y,y)) are present, but ((z,z)) is missing. Step 3: Add the missing self-pair of the missing element.

Step 1: A relation on (A) is always a subset of (A \times A). Step 2: A reflexive relation is also a relation, so it is a subset of (A \times A). Step 3: First recall the basic definition of a relation.

Step 1: The smallest reflexive relation contains only diagonal pairs. Step 2: (A) has 5 elements, so 5 self-pairs are required. Step 3: For minimum reflexive relation, the number is (n).