Reflexive relation

Step 1: Every reflexive relation contains all self-pairs. Step 2: Since (2 \in A), ((2,2)) must be present in every reflexive relation. Step 3: Pairs with different elements are not always required.

Step 1: Reflexivity makes only self-pairs compulsory. Step 2: ((1,2)) is not a self-pair, so it is not necessary. Step 3: Pay attention to the phrase not necessary in the question.

Step 1: The identity relation contains only self-pairs. Step 2: (A) has 4 elements, so it has 4 pairs. Step 3: The identity relation and minimum reflexive relation have the same count.

Step 1: The given (R) has all self-pairs. Step 2: It has no extra pair, so it is the minimum reflexive relation. Step 3: The minimum reflexive relation is also the identity relation.

Step 1: Identify the required self-pairs separately. Step 2: ((1,1),(2,2),(3,3)) are present, so (R) is reflexive. Step 3: Do not change the answer because of extra cyclic pairs.

Step 1: All self-pairs of (A) are required. Step 2: For (3), ((3,3)) is necessary but not given. Step 3: Even if other pairs are present, one missing self-pair fails reflexivity.

Step 1: Reflexive relation is identified by self-pairs. Step 2: So the first task is to check ((a,a)) for every element. Step 3: Check symmetric or other properties later.

Step 1: In a reflexive relation, each element of (A) is related to itself. Step 2: Since (4 \in A), ((4,4)) must be present. Step 3: Do not treat pairs with different elements as compulsory.

Step 1: Not reflexive means at least one required self-pair is missing. Step 2: So for some (a \in A), ((a,a) \notin R). Step 3: Understanding the opposite condition helps identify the property quickly.

Step 1: To test reflexivity, put ((a,a)) in the rule. Step 2: For ((a,a)), (a=b) is true, so every self-pair is included. Step 3: In an or condition, one true condition is enough.

Step 1: In a reflexive relation every element must be related to itself. Step 2: So for (1,2,3), the required pairs are ((1,1),(2,2),(3,3)). Step 3: In exams, check diagonal pairs first.

Step 1: For elements (a) and (b), the pairs ((a,a)) and ((b,b)) are required. Step 2: Both pairs are present in (R), so the extra pair does not matter. Step 3: Reflexivity only needs all self-pairs.

Step 1: A reflexive relation must contain the self-pair of every element of (A). Step 2: For (6), the pair ((6,6)) is missing. Step 3: If even one required pair is absent, the relation is not reflexive.

Step 1: The smallest reflexive relation contains only the required self-pairs. Step 2: For (5) and (7), the needed pairs are ((5,5)) and ((7,7)). Step 3: Extra pairs are not compulsory.

Step 1: (A\times A) contains all possible ordered pairs from (A). Step 2: It includes ((1,1)) and ((2,2)), so it is reflexive. Step 3: The universal relation is always reflexive.

Step 1: A non-empty set has at least one element. Step 2: The empty relation has no self-pairs. Step 3: Therefore, it is not reflexive on a non-empty set.

Step 1: A reflexive relation needs one self-pair for each element. Step 2: (A) has (4) elements, so at least (4) pairs are needed. Step 3: The minimum count equals the number of elements.

Step 1: For (x,y,z), all three self-pairs ((x,x),(y,y),(z,z)) are required. Step 2: Only the second relation contains all three. Step 3: Extra pairs do not break reflexivity.

Step 1: The condition (a=b) gives self-pairs. Step 2: It gives ((1,1),(3,3),(5,5)). Step 3: Equality is a simple example of a reflexive relation.

Step 1: Reflexivity needs ((a,a)) for every (a\in A). Step 2: Every number is less than or equal to itself, so (a\le a) is true. Step 3: The relation (\le) is generally reflexive.