Reflexive relation

Correct answer: A. (R) is reflexive

Step 1: Put ((a,a)), giving (a^2+a^2\geq 2a^2). Step 2: This is (2a^2\geq 2a^2), true for every element. Step 3: Also, (a^2+b^2-2ab=(a-b)^2), which is never negative.

Correct answer: B. Not reflexive

Step 1: For ((a,a)), ((a-a)^2=0). Step 2: (0>0) is false, so no diagonal pair belongs. Step 3: ((a-b)^2>0) means (a\neq b), so the relation is not reflexive.

Correct answer: A. Reflexive

Step 1: The square of any real number is non-negative. Step 2: For ((a,a)), the value is (0), and (0\geq 0) is true. Step 3: In relations involving non-negative squares, check diagonal pairs carefully.

Correct answer: A. Yes

Step 1: Put (b=a) to test reflexivity. Step 2: Since every (a \in A) is non-zero, (\frac{a}{a}=1) is true. Step 3: In fraction-based relations, also check that the denominator is not zero.

Correct answer: B. No because ((0,0)) is not defined

Step 1: Reflexivity needs ((0,0)), ((1,1)), and ((2,2)). Step 2: (\frac{0}{0}) is undefined, so ((0,0)) cannot belong to the relation. Step 3: With sets containing zero, read division conditions carefully.

Correct answer: A. Because every (a) divides itself

Step 1: Test the condition on ((a,a)). Step 2: For every (a), both (a\mid a) and (a\mid a) are true. Step 3: A number divides itself, provided it is non-zero.

Correct answer: A. Yes

Step 1: For ((a,a)), (\frac{a}{a}=1). Step 2: Since (1) is an integer, every diagonal pair is included. Step 3: In quotient conditions, dividing a non-zero element by itself gives (1).

Correct answer: A. Because every number has the same parity as itself

Step 1: In ((a,a)), both entries are the same number. Step 2: A number has the same parity as itself, so every diagonal pair belongs. Step 3: Relations based on the same property are usually reflexive.

Correct answer: B. No

Step 1: In ((a,a)), both numbers are identical. Step 2: A number cannot have parity opposite to itself, so no diagonal pair belongs. Step 3: Relations based on opposite properties are not reflexive.

Correct answer: A. (R) is reflexive

Step 1: Reflexivity requires the presence of all diagonal pairs. Step 2: Here ((1,1),(2,2),(3,3)) are all present, so the relation is reflexive. Step 3: Extra pairs do not destroy reflexivity.

Correct answer: A. Add ((4,4))

Step 1: Four diagonal pairs are needed for (A). Step 2: The relation has ((1,1),(2,2),(3,3)), but not ((4,4)). Step 3: For minimum addition, add only the missing diagonal pair.

Correct answer: B. (R) is not reflexive

Step 1: A reflexive relation on an (n)-element set needs all (n) diagonal pairs. Step 2: Here only (n-1) diagonal pairs are present, so one element is not related to itself. Step 3: Missing even one diagonal pair destroys reflexivity.

Correct answer: A. 5

Step 1: A reflexive relation must contain one diagonal pair for each element. Step 2: For 5 elements, the minimum number of required pairs is 5. Step 3: For minimum count, count only compulsory diagonal pairs.

Correct answer: B. (2^{12})

Step 1: Total ordered pairs are (4^2=16). Step 2: A reflexive relation must include 4 diagonal pairs, so the remaining (16-4=12) pairs are optional. Step 3: If (m) pairs are optional, the number of choices is (2^m).

Correct answer: C. (2^{n^2-n})

Step 1: There are (n^2) total ordered pairs. Step 2: The (n) diagonal pairs are compulsory for reflexivity. Step 3: The remaining (n^2-n) pairs are optional, so the count is (2^{n^2-n}).

Correct answer: C. 448

Step 1: Total relations are (2^{3^2}=2^9=512). Step 2: Reflexive relations are (2^{9-3}=2^6=64). Step 3: Non-reflexive relations are (512-64=448).

Correct answer: B. 6

Step 1: Reflexivity fixes the 3 diagonal pairs. Step 2: To have exactly 5 pairs, choose 2 more from the 6 non-diagonal pairs. Step 3: The count is (\binom{6}{2}=15), so none of the listed numerical choices is correct.

Correct answer: C. 15

Step 1: The 3 diagonal pairs are compulsory. Step 2: Since exactly 5 pairs are needed, choose 2 additional pairs from (3^2-3=6) non-diagonal pairs. Step 3: Therefore the number is (\binom{6}{2}=15).

Correct answer: C. 66

Step 1: The 4 diagonal pairs are compulsory. Step 2: To have exactly 6 pairs, choose 2 extra pairs from (16-4=12) non-diagonal pairs. Step 3: The number is (\binom{12}{2}=66).

Correct answer: D. 16

Step 1: Reflexivity only requires diagonal pairs to be present. Step 2: All other pairs may also be included, so the maximum case is the universal relation. Step 3: A 4-element set has (4^2=16) ordered pairs.

Correct answer: B. Not reflexive

Step 1: (A\times A) contains all pairs, but ((2,2)) has been removed. Step 2: Reflexivity requires ((2,2)) as well. Step 3: Removing even one diagonal pair from the universal relation destroys reflexivity.

Correct answer: A. Yes

Step 1: (A\times A) contains all diagonal pairs. Step 2: The removed pairs ((1,2)) and ((3,4)) are not diagonal pairs. Step 3: Removing non-diagonal pairs does not affect reflexivity.

Correct answer: A. (I) is reflexive

Step 1: The identity relation contains each element paired with itself. Step 2: All three diagonal pairs of (A) are present in (I). Step 3: The identity relation is the smallest reflexive relation on a set.

Correct answer: B. No

Step 1: A non-empty set has at least one element (a). Step 2: Reflexivity requires ((a,a)), but the empty relation has no pairs. Step 3: The empty relation is not reflexive on a non-empty set.

Correct answer: A. It is reflexive because there is no element to check

Step 1: Reflexivity says that for every (a \in A), ((a,a)\in R). Step 2: When (A=\phi), there is no element (a) for which the condition can fail. Step 3: On the empty set, the condition is treated as true by vacuous truth.