Reflexive relation

Correct answer: A. Yes, because (a=b) includes every ((a,a))

Step 1: To test reflexivity, put ((a,a)) in the rule. Step 2: For ((a,a)), (a=b) is true, so every self-pair is included. Step 3: In an or condition, one true condition is enough.

Correct answer: B. Only ((1,1),(2,2),(3,3))

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

Correct answer: A. Reflexive

Step 1: For elements (a) and (b), the pairs ((a,a)) and ((b,b)) are required. Step 2: Both pairs are present in (R), so the extra pair does not matter. Step 3: Reflexivity only needs all self-pairs.

Correct answer: C. Because ((6,6)) is missing

Step 1: A reflexive relation must contain the self-pair of every element of (A). Step 2: For (6), the pair ((6,6)) is missing. Step 3: If even one required pair is absent, the relation is not reflexive.

Correct answer: B. (R={(5,5),(7,7)})

Step 1: The smallest reflexive relation contains only the required self-pairs. Step 2: For (5) and (7), the needed pairs are ((5,5)) and ((7,7)). Step 3: Extra pairs are not compulsory.

Correct answer: A. (R) is reflexive

Step 1: (A\times A) contains all possible ordered pairs from (A). Step 2: It includes ((1,1)) and ((2,2)), so it is reflexive. Step 3: The universal relation is always reflexive.

Correct answer: B. It is never reflexive

Step 1: A non-empty set has at least one element. Step 2: The empty relation has no self-pairs. Step 3: Therefore, it is not reflexive on a non-empty set.

Correct answer: C. (4)

Step 1: A reflexive relation needs one self-pair for each element. Step 2: (A) has (4) elements, so at least (4) pairs are needed. Step 3: The minimum count equals the number of elements.

Correct answer: B. ({(x,x),(y,y),(z,z),(x,y)})

Step 1: For (x,y,z), all three self-pairs ((x,x),(y,y),(z,z)) are required. Step 2: Only the second relation contains all three. Step 3: Extra pairs do not break reflexivity.

Correct answer: A. Reflexive

Step 1: The condition (a=b) gives self-pairs. Step 2: It gives ((1,1),(3,3),(5,5)). Step 3: Equality is a simple example of a reflexive relation.

Correct answer: A. Yes, because (a\le a) for every (a)

Step 1: Reflexivity needs ((a,a)) for every (a\in A). Step 2: Every number is less than or equal to itself, so (a\le a) is true. Step 3: The relation (\le) is generally reflexive.

Correct answer: B. Not reflexive

Step 1: Reflexivity needs ((1,1),(2,2),(3,3)). Step 2: No number satisfies (a<a). Step 3: The relation (<) is not reflexive.

Correct answer: A. ((x,x)\in R)

Step 1: The key idea of a reflexive relation is being related to itself. Step 2: This means ((x,x)\in R) for every (x). Step 3: Remembering the definition helps solve such questions quickly.

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

Step 1: For (0) and (1), the pairs ((0,0)) and ((1,1)) are required. Step 2: ((0,0)) is already present, but ((1,1)) is missing. Step 3: Add the missing diagonal pair.

Correct answer: C. ((r,r))

Step 1: In a reflexive relation, every element is related to itself. Step 2: Since (r) is in (A), ((r,r)) must be present. Step 3: Pairs with different elements are not compulsory.

Correct answer: A. Smallest reflexive relation

Step 1: It contains the self-pair of every element. Step 2: It has no extra pairs, so it is the smallest reflexive relation. Step 3: This is also the identity relation on the set.

Correct answer: A. Whether all ((a,a)) pairs are present

Step 1: Reflexivity is checked through self-pairs. Step 2: For every element (a), ((a,a)) must be present. Step 3: In exams, list the diagonal pairs directly.

Correct answer: A. (R) is reflexive

Step 1: The required pairs for (A) are ((1,1),(2,2),(3,3)). Step 2: All three are present in (R). Step 3: Other pairs do not affect reflexivity.

Correct answer: B. ((2,2))

Step 1: The elements (1,2,3) need their self-pairs. Step 2: ((1,1)) and ((3,3)) are present, but ((2,2)) is missing. Step 3: Focus on the missing self-pair.

Correct answer: A. Because (a-a=0) for every (a)

Step 1: For reflexivity, test an element with itself. Step 2: For every (a), (a-a=0), so ((a,a)\in R). Step 3: Put (b=a) in the condition to check quickly.

Correct answer: A. Yes, because (a+a=2a) is even

Step 1: For reflexivity, put ((a,a)) into the condition. Step 2: (a+a=2a) is always even, so every ((a,a)) is included. Step 3: Checking the self-pair condition is the fastest method.

Correct answer: B. Not reflexive

Step 1: For ((a,a)), the sum is (a+a=2a). Step 2: (2a) is always even, not odd. Step 3: So no self-pair satisfies the condition.

Correct answer: A. Yes, because every (a) divides itself

Step 1: Reflexivity needs ((a,a)) for every (a). Step 2: Every non-zero number divides itself, so ((a,a)) is included. Step 3: In divisibility questions, remember (a\mid a) first.

Correct answer: A. Because (a=a+1) is never true

Step 1: For reflexivity, putting (b=a) would require (a=a+1). Step 2: This is not true for any number. Step 3: A condition that gives no self-pair is not reflexive.

Correct answer: B. ({(10,10)})

Step 1: The singleton set has only (10). Step 2: Reflexivity requires ((10,10)). Step 3: The definition stays the same even for small sets.