Reflexive relation

Step 1: The only necessary reflexive pairs are of the form ((a,a)). Step 2: For (1,2,3), these are ((1,1),(2,2),(3,3)). Step 3: If the question says only necessary, do not choose extra pairs.

Step 1: The condition of reflexivity applies to every element. Step 2: So if (a\in A), then ((a,a)\in R) must hold. Step 3: In definition questions, connect the words with the symbolic form.

Step 1: For (0,1,2), the required pairs are ((0,0),(1,1),(2,2)). Step 2: All three are present in the relation. Step 3: The extra pair ((0,1)) does not affect reflexivity.

Step 1: (0) is also an element of the set. Step 2: Reflexivity requires ((0,0)). Step 3: Add the missing self-pair to make the relation reflexive.

Step 1: Pairs only of the form ((a,a)) make the identity relation. Step 2: Here every element has its self-pair. Step 3: The identity relation is reflexive, but every reflexive relation need not be only identity.

Step 1: The identity relation contains only self-pairs. Step 2: The first option gives exactly those pairs. Step 3: If an extra pair is added, the relation may remain reflexive but is no longer identity.

Step 1: The main condition of reflexivity is the presence of all ((a,a)) pairs. Step 2: If even one is missing, the condition fails. Step 3: To identify a non-reflexive relation, look for a missing diagonal pair.

Step 1: For reflexivity, checking (a=a) is enough. Step 2: The condition includes (a=b), so every ((a,a)) is included. Step 3: If equality is included in the condition, check reflexivity immediately.

Step 1: Reflexivity needs pairs where (a=a). Step 2: But the condition (a\ne b) excludes all self-pairs. Step 3: Relations based on inequality are usually not reflexive.

Step 1: Every element being related to itself is the sign of reflexivity. Step 2: Symbolically, ((a,a)\in R) for every (a\in A). Step 3: Remember the definition by connecting it with simple classroom language.

Step 1: (A) has 3 elements, so (A \times A) has 9 pairs. Step 2: A reflexive relation must contain 3 self-pairs, while the remaining (9-3=6) pairs are optional. Step 3: Hence the number of reflexive relations is (2^6=64).

Step 1: For 4 elements, (A \times A) has 16 pairs. Step 2: Reflexivity fixes 4 diagonal pairs, so (16-4=12) pairs are optional. Step 3: Each optional pair has two choices, giving (2^{12}).

Step 1: (A \times A) contains (n^2) pairs. Step 2: In a reflexive relation, (n) self-pairs are compulsory, so (n^2-n) pairs are free. Step 3: Therefore the number is (2^{n^2-n}).

Step 1: For (A), the required pairs are ((1,1),(2,2),(3,3),(4,4)). Step 2: Only ((3,3)) is missing in the given relation. Step 3: So adding one pair is enough.

Step 1: A reflexive relation needs self-pairs for (a,b,c,d). Step 2: ((a,a)) and ((c,c)) are present, but ((b,b)) and ((d,d)) are missing. Step 3: Identifying missing diagonal pairs is the key.

Step 1: To test reflexivity, put ((a,a)) in the rule. Step 2: Every number is equal to itself, so (a\le a) is true. Step 3: In a (\le) relation, the equality part gives reflexivity.

Step 1: Reflexivity needs ((1,1),(2,2),(3,3)). Step 2: Under (a<b), (1<1) is false, so ((1,1)) is not formed. Step 3: Strict inequality does not include self-pairs.

Step 1: For reflexivity, put (b=a). Step 2: Then (a-b=a-a=0), and (0) is divisible by (3). Step 3: In difference-based relations, quickly test (a-a).

Step 1: Reflexivity needs ((a,a)) for every (a). Step 2: For ((a,a)), the sum is (a+a=2a), which is even. Step 3: If the rule asks for an odd sum, self-pairs are not included.

Step 1: Reflexivity compares a number with itself. Step 2: Any number has the same remainder as itself when divided by (2). Step 3: Treat (a \equiv a \pmod{2}) as the basic test in congruence relations.