Reflexive relation

Step 1: In a diagonal pair, both elements are the same. Step 2: Therefore (|a-a|=0), and (0) is even. Step 3: Hence every ((a,a)) belongs to the relation.

Step 1: In diagonal pairs, (a=b). Step 2: The condition also requires (a\neq b), so no diagonal pair can be in (R). Step 3: For four elements, all four diagonal pairs must be added.

Step 1: On the diagonal, (a=b) is always true. Step 2: In an or condition, one true part is enough. Step 3: Therefore every ((a,a)) belongs to (R), so it is reflexive.

Step 1: On the diagonal, (a=b) is true. Step 2: But (a+b=2a) is always even, not odd. Step 3: In an and condition, both parts must be true, so the relation is not reflexive.

Step 1: On the diagonal, (a\neq b) is false. Step 2: But (a+b=2a) is even, so the second part is true. Step 3: In an or condition, one true part includes every diagonal pair.

Step 1: Reflexivity needs ((1,1),(2,2),(3,3)). Step 2: The relation has ((3,3)), but misses ((1,1)) and ((2,2)). Step 3: Therefore (2) pairs must be added.

Step 1: On a four-element set, (4) diagonal pairs are compulsory. Step 2: To have exactly (7) pairs, choose (3) non-diagonal pairs. Step 3: There are (16-4=12) non-diagonal pairs, so the number is (\binom{12}{3}).

Step 1: On a five-element set, the minimum reflexive relation has (5) diagonal pairs. Step 2: Having (6) more pairs means choosing (6) non-diagonal pairs. Step 3: There are (25-5=20) non-diagonal pairs, so the count is (\binom{20}{6}).

Step 1: A reflexive relation must contain (n) diagonal pairs. Step 2: To have exactly (n+2) pairs, choose (2) non-diagonal pairs. Step 3: Since there are (n^2-n) non-diagonal pairs, the number is (\binom{n^2-n}{2}).

Step 1: On a three-element set, the number of reflexive relations is (2^{9-3}=2^6=64). Step 2: This count includes the universal relation. Step 3: Excluding the universal relation gives (64-1=63).

Step 1: On a four-element set, the number of reflexive relations is (2^{16-4}=2^{12}). Step 2: The identity relation is one of them. Step 3: Removing it leaves (2^{12}-1) relations.

Step 1: The (4) diagonal pairs are compulsory. Step 2: Among the (12) non-diagonal pairs, ((1,2)) is compulsory and ((2,1)) is forbidden. Step 3: The remaining (10) non-diagonal pairs are optional, giving (2^{10}) relations.

Step 1: On a three-element set, there are (2^6=64) reflexive relations. Step 2: The only one with no non-diagonal pair is the identity relation. Step 3: Hence the number with at least one non-diagonal pair is (64-1=63).

Step 1: The four diagonal pairs are compulsory. Step 2: To have (9) total pairs, choose (5) non-diagonal pairs. Step 3: Choosing (5) from (12) non-diagonal pairs gives (\binom{12}{5}).

Step 1: A reflexive relation must contain all five diagonal pairs. Step 2: If no non-diagonal pair is allowed, the relation is exactly the identity relation. Step 3: Therefore only (1) such relation exists.

Step 1: (A\times A) has (16) pairs. Step 2: Exactly two non-diagonal pairs are missing while all diagonal pairs are present. Step 3: Hence the total number of pairs is (16-2=14).

Step 1: (\mathcal{P}(S)) has (2^3=8) elements. Step 2: Every set is a subset of itself, so every ((X,X)) is in the relation. Step 3: Therefore the number of diagonal pairs is (8).

Step 1: (A) has (8) sets, so (8) diagonal pairs are needed. Step 2: On the diagonal, (X\cap X=X), which is empty only when (X=\varnothing). Step 3: One diagonal pair is already present, so (8-1=7) must be added.

Step 1: For reflexivity, put (Y=X). Step 2: Then (X\cup X=X), which is true for every set. Step 3: Hence every ((X,X)) belongs to the relation.

Step 1: On the diagonal, (X\setminus X=\varnothing). Step 2: Removing a set from itself gives the empty set. Step 3: Therefore all diagonal pairs are in the relation.