Reflexive relation

Step 1: Reflexivity fails when a required self-pair is absent. Step 2: If ((a,a)\notin R) for some (a\in A), then (R) is not reflexive. Step 3: To find the failure, check each diagonal pair.

Step 1: Each element of (A) needs its self-pair. Step 2: ((1,1),(2,2),(3,3)) are all in (R). Step 3: Reflexivity is decided by these pairs.

Step 1: In the identity relation, each element is related to itself. Step 2: So ((a,a)) is present for every (a\in A). Step 3: The identity relation is a basic example of a reflexive relation.

Step 1: The set (A) has three elements (1,2,3). Step 2: ((3,3)) is also required, but it is absent. Step 3: If one element misses its self-pair, the relation is not reflexive.

Step 1: For ((a,a)), we get (a-a=0). Step 2: (0) is even, so every self-pair satisfies the condition. Step 3: For subtraction conditions, test by putting (a=a).

Step 1: For a self-pair, (a-b=a-a=0). Step 2: (0) is not odd, so ((a,a)) does not satisfy the condition. Step 3: Therefore the relation is not reflexive.

Step 1: For reflexivity, check (a\ge a). Step 2: Every number is equal to itself, so (a\ge a) is true. Step 3: The relation (\ge) is reflexive.

Step 1: For a self-pair, put (b=a) in (a>b). Step 2: This gives (a>a), which is false. Step 3: Hence the relation (>) is not reflexive.

Step 1: The four elements of (A) need their self-pairs. Step 2: ((1,1),(2,2),(3,3),(4,4)) are all given. Step 3: ((1,4)) is extra and does not affect reflexivity.

Step 1: The smallest reflexive relation contains only self-pairs. Step 2: For (n) elements, there are (n) such pairs. Step 3: The minimum number of pairs is always (n).

Step 1: The universal relation is (A\times A). Step 2: With (n) elements, it has (n^2) ordered pairs. Step 3: It is also reflexive because it contains all self-pairs.

Step 1: For (A={1,2}), ((1,1)) and ((2,2)) are required. Step 2: The third relation does not contain ((2,2)). Step 3: Therefore it is not reflexive.

Step 1: The condition (a=b) gives every self-pair. Step 2: ((1,1),(2,2),(3,3)) will all be in (R). Step 3: The other condition may add pairs, but reflexivity remains.

Step 1: In a self-pair, (a=b). Step 2: The condition (a\ne b) excludes all self-pairs. Step 3: Therefore the relation is not reflexive.

Step 1: If a relation is reflexive, every element is related to itself. Step 2: For (4,5,6), the pairs ((4,4),(5,5),(6,6)) must be present. Step 3: Reflexivity alone does not decide other pairs.

Step 1: (R) is already reflexive because all self-pairs are present. Step 2: Adding any extra pair does not remove those pairs. Step 3: Reflexivity is not spoiled by extra pairs.

Step 1: For (1,2,3), the pairs ((1,1),(2,2),(3,3)) are needed. Step 2: The first two are present, but ((3,3)) is missing. Step 3: Add this pair to make the relation reflexive.

Step 1: There is one self-pair for each element. Step 2: (A) has (4) elements, so (4) such pairs are required. Step 3: When counting, directly look at the number of elements.

Step 1: Reflexivity does not require all possible pairs. Step 2: It only requires the self-pair of every element. Step 3: So the relation can be reflexive even if some other pairs are missing.

Step 1: Since (R) is reflexive, it has all self-pairs. Step 2: (S) contains every pair of (R), so those self-pairs are also in (S). Step 3: A larger relation containing a reflexive relation remains reflexive.