Reflexive relation

Correct answer: A. (\binom{12}{3})

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}).

Correct answer: A. (\binom{20}{6})

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}).

Correct answer: A. (\binom{n^2-n}{2})

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}).

Correct answer: A. (63)

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).

Correct answer: A. (2^{12}-1)

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.

Correct answer: A. (2^{10})

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.

Correct answer: A. (63)

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).

Correct answer: A. (\binom{12}{5})

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}).

Correct answer: A. 1

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.

Correct answer: B. 14

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).

Correct answer: C. 8

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).

Correct answer: B. 7

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.

Correct answer: A. Yes

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.

Correct answer: A. It is reflexive

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.

Correct answer: A. Reflexive

Step 1: On the diagonal, (X=Y) is true. Step 2: The or condition includes every ((X,X)) through the equality part. Step 3: Therefore the relation is reflexive.

Correct answer: A. 1

Step 1: On the diagonal, (X\triangle X=\varnothing). Step 2: The condition (X\triangle X=X) holds only when (X=\varnothing). Step 3: Hence only ((\varnothing,\varnothing)) is a diagonal pair.

Correct answer: B. No

Step 1: Reflexivity also requires ((1,1)). Step 2: The number (1) has no factor greater than (1). Step 3: Missing even one diagonal pair makes the relation non-reflexive.

Correct answer: A. Reflexive

Step 1: Every (a\in A) is divisible by some power of (2). Step 2: In ((a,a)), both entries are the same, so they share that same power. Step 3: Therefore all diagonal pairs are present.

Correct answer: A. Yes

Step 1: For reflexivity, put (a=b). Step 2: Then the ratio is (\frac{a}{a}=1=2^0), an integral power of (2). Step 3: Hence every ((a,a)) belongs to the relation.

Correct answer: C. 4

Step 1: On the diagonal, the condition becomes (2a\leq a^2). Step 2: Since (a>0), this becomes (2\leq a). Step 3: (a=2,3,4,5) work, so there are (4) diagonal pairs.

Correct answer: B. 1

Step 1: On the diagonal, the condition is (2a\leq a^2). Step 2: This fails for (a=1) and holds for (a=2,3,4,5). Step 3: Therefore only ((1,1)) must be added.

Correct answer: A. It is reflexive

Step 1: On the diagonal, (a^3-a^3=0). Step 2: This is true for every (a\in A). Step 3: Therefore all diagonal pairs are in the relation, so (R) is reflexive.

Correct answer: B. 2

Step 1: On the diagonal, (2a^2=2a). Step 2: This becomes (a^2=a), i.e. (a(a-1)=0). Step 3: (a=0,1) work, so there are (2) diagonal pairs.

Correct answer: B. 3

Step 1: A reflexive relation on this five-element set needs five diagonal pairs. Step 2: The condition gives diagonal pairs only for (a=0) and (a=1). Step 3: Hence (5-2=3) diagonal pairs must be added.

Correct answer: A. For every (a), (a-a=0) and (0) is divisible by (6)

Step 1: For reflexivity, check ((a,a)). Step 2: Then (a-a=0), and (0) is divisible by (6). Step 3: In difference-divisibility relations, the diagonal difference is always zero.