Reflexive relation

Step 1: Reflexivity only requires ((1,1),(2,2),(3,3)). Step 2: ((2,3)) is not a self-pair. Step 3: In such questions, distinguish ((a,a)) from ((a,b)).

Option C is correct. A set with two elements has 2×2=4 possible ordered pairs, so it has 2^4=16 total relations. For a relation to be reflexive, both diagonal pairs (1,1) and (2,2) must be included; the other two pairs may be chosen freely, giving 2^2=4 reflexive relations. Therefore, the number that are not reflexive is 16−4=12.

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).

Reflexivity requires every element to be related to itself. For A={a1,a2,...,an}, the compulsory pairs are (a1,a1), (a2,a2), ..., (an,an), giving exactly n diagonal ordered pairs. All off-diagonal pairs may be chosen independently, but they are not compulsory. Therefore option A is correct; n^2 counts all possible positions in a relation, not the required diagonal pairs.

Step 1: The three elements of (A) need three diagonal pairs. Step 2: The third relation does not contain ((3,3)). Step 3: In options, look for the missing diagonal pair.

Step 1: The condition (a=b) gives pairs only of the form ((a,a)). Step 2: Every element is related to itself, so it is the identity relation. Step 3: The identity relation is also reflexive.

Step 1: Reflexivity needs (a\le a) for every (a). Step 2: Every number is equal to itself, so (a\le a) is true. Step 3: Relations with (\le) are easy examples of reflexive relations.

Step 1: Reflexivity would require (a<a) for every (a). Step 2: No number is less than itself. Step 3: A relation based on (<) is generally not reflexive.

Step 1: For reflexivity, check (a-a). Step 2: (a-a=0), and (0) is even. Step 3: If the condition is true for every element with itself, the relation is reflexive.

Step 1: For reflexivity, check (a+a). Step 2: (a+a=2a) is always even, not odd. Step 3: If ((a,a)) does not satisfy the condition, the relation is not reflexive.

A relation on A is reflexive when (a, a) belongs to R for every a in A. Here, 1 divides 1, 2 divides 2 and 3 divides 3, because every non-zero integer divides itself. Thus (1,1), (2,2) and (3,3) are all in R, so option A is correct. Options C and D check only part of the required condition.

Step 1: Check ((1,1),(2,2),(3,3)) for reflexivity. Step 2: (1+1=2) and (3+3=6), so not all self-pairs satisfy the condition. Step 3: Missing even one required self-pair means the relation is not reflexive.

Step 1: (|a-b|=0) holds exactly when (a=b). Step 2: So ((1,1),(2,2),(3,3)) are all included. Step 3: Equality-based relations are good examples of reflexivity.

Step 1: A reflexive relation relates every element to itself. Step 2: For (p,q,r), the pairs ((p,p),(q,q),(r,r)) are required. Step 3: Even if symbols change, the rule stays the same.

Step 1: Since (4\in A), reflexivity requires ((4,4)). Step 2: This pair is missing, so the condition fails. Step 3: In checking reflexivity, any missing diagonal pair gives the answer quickly.

Step 1: Reflexivity requires only all pairs of the form ((a,a)). Step 2: Other types of pairs are not compulsory. Step 3: Do not confuse missing extra pairs with missing reflexive pairs.

Step 1: First check the diagonal pairs. Step 2: ((1,1),(2,2),(3,3)) are all present. Step 3: Whatever the other pairs are, these three make the relation reflexive.

Step 1: A reflexive relation does not require all possible pairs. Step 2: It only requires every ((a,a)) pair. Step 3: A relation with all pairs is universal, but reflexivity itself needs only self-pairs.

Step 1: The identity relation contains only ((a,a)) for each element. Step 2: The set (A) has five elements, so there are five pairs. Step 3: The number of pairs in the identity relation equals the number of elements.

Step 1: For (A={1,2}), ((1,1)) and ((2,2)) are required. Step 2: Both pairs are present. Step 3: This relation is also (A\times A), so it is reflexive.