Reflexive relation

Correct answer: A. (S) will also be reflexive

Step 1: Since (R) is reflexive, it has all self-pairs. Step 2: (S) contains every pair of (R), so those self-pairs are also in (S). Step 3: A larger relation containing a reflexive relation remains reflexive.

Correct answer: A. ( (1,1) )

Step 1: In a reflexive relation, every element is related to itself. Step 2: So for element (1), the pair ( (1,1) ) must be present. Step 3: In exams, first check the diagonal pairs.

Correct answer: B. (2)

Step 1: A reflexive relation must contain ((x,x)) for every element. Step 2: Here there are two elements, so ((a,a)) and ((b,b)) are required. Step 3: The minimum number of pairs equals the number of elements in the set.

Correct answer: C. (4)

Step 1: In a reflexive relation, ((x,x)) must exist for every element (x). Step 2: The set has four elements, so four compulsory pairs are needed. Step 3: For such questions, count the number of elements directly.

Correct answer: C. (R={(1,1),(2,2)})

Step 1: For (A={1,2}), both ((1,1)) and ((2,2)) are required. Step 2: Only the third relation contains both. Step 3: Extra pairs may exist, but self-pairs must not be missing.

Correct answer: A. Reflexive

Step 1: To be reflexive, ((1,1)) and ((2,2)) are required. Step 2: The given relation contains both pairs. Step 3: The extra pair ((1,2)) does not affect reflexivity.

Correct answer: C. Because ((3,3)) is missing

Step 1: The set (A) also contains element (3). Step 2: A reflexive relation must contain ((3,3)). Step 3: Missing even one diagonal pair makes the relation non-reflexive.

Correct answer: A. (I_A={(1,1),(2,2),(3,3)})

Step 1: An identity relation contains only pairs of the form ((x,x)). Step 2: For (1,2,3), the pairs are ((1,1),(2,2),(3,3)). Step 3: The identity relation is always reflexive.

Correct answer: A. Yes

Step 1: The set has only one element (5). Step 2: The required pair ((5,5)) is present. Step 3: Reflexivity on a singleton set is very easy to check.

Correct answer: B. No

Step 1: A reflexive relation needs ((1,1)) and ((2,2)). Step 2: The empty relation has no ordered pair. Step 3: On a non-empty set, the empty relation is not reflexive.

Correct answer: A. For every (a\in A), ((a,a)\in R)

Step 1: Reflexivity is based on every element being related to itself. Step 2: Therefore ((a,a)) must be present for every (a). Step 3: Remember ((a,a)) as the key sign of reflexivity.

Correct answer: D. ((x,y))

Step 1: In a reflexive relation, only self-pairs are compulsory. Step 2: ((x,x),(y,y),(z,z)) are required, but ((x,y)) is not. Step 3: Distinguish between compulsory and extra pairs.

Correct answer: A. (R) is reflexive

Step 1: (A\times A) contains all possible ordered pairs from (A). Step 2: So ((1,1),(2,2),(3,3)) are included. Step 3: The universal relation is always reflexive.

Correct answer: C. (n)

Step 1: Each element needs one pair of the form ((a,a)). Step 2: For (n) elements, there are (n) such pairs. Step 3: If the minimum is asked, do not count extra pairs.

Correct answer: A. Reflexive

Step 1: Reflexivity needs all three diagonal pairs. Step 2: ((1,1),(2,2),(3,3)) are all present. Step 3: ((1,2)) is extra, but it does not change reflexivity.

Correct answer: C. ((3,3)) is missing

Step 1: Element (3) is also in the set. Step 2: Reflexivity requires ((3,3)). Step 3: Check diagonal pairs before looking at non-diagonal pairs.

Correct answer: A. Reflexive

Step 1: (R) contains ((a,a)) for every (a\in A). Step 2: That is exactly the basic condition for reflexivity. Step 3: This form also represents the identity relation.

Correct answer: A. A loop at every point

Step 1: In a reflexive relation, each element is related to itself. Step 2: In a graph, this appears as a loop at every point. Step 3: In diagram questions, check the loop at every point.

Correct answer: A. All main diagonal entries are (1)

Step 1: In a matrix, pairs of the form ((a,a)) lie on the main diagonal. Step 2: For reflexivity, each such entry must be (1). Step 3: In matrix questions, inspect the main diagonal first.

Correct answer: A. It is reflexive

Step 1: The main diagonal represents pairs of the form ((a,a)). Step 2: All diagonal entries are (1), so all self-pairs are present. Step 3: The fastest matrix check for reflexivity is the diagonal.

Correct answer: C. (3)

Step 1: Count the ordered pairs in the relation. Step 2: ((1,1),(2,2),(3,3)) make three pairs. Step 3: In an identity relation, the number of pairs equals the number of elements.

Correct answer: D. ((2,3))

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)).

Correct answer: D. (64)

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).

Correct answer: A. n

Direct answer: A, n. Let A={a1,a2,...,an}. Reflexivity means every element is related to itself, so the compulsory pairs are (a1,a1), (a2,a2), ..., (an,an). There is one such diagonal pair for each of the n elements, hence exactly n required pairs. The remaining n(n−1) off-diagonal pairs may be included or omitted independently. Option B, n², counts every possible ordered position in A×A, not only the compulsory ones. Option C counts off-diagonal positions, which are optional rather than required. Option D is an exponential count and does not count the required diagonal pairs. Memory cue: “reflexive = all self-pairs”; count the elements, not all pair positions.

Correct answer: C. (R={(1,1),(2,2),(1,3)})

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.