Reflexive relation

Correct answer: A. Identity relation and reflexive

Step 1: The condition (a=b) gives pairs only of the form ((a,a)). Step 2: Every element is related to itself, so it is the identity relation. Step 3: The identity relation is also reflexive.

Correct answer: A. Yes

Step 1: Reflexivity needs (a\le a) for every (a). Step 2: Every number is equal to itself, so (a\le a) is true. Step 3: Relations with (\le) are easy examples of reflexive relations.

Correct answer: B. No

Step 1: Reflexivity would require (a<a) for every (a). Step 2: No number is less than itself. Step 3: A relation based on (<) is generally not reflexive.

Correct answer: A. Yes

Step 1: For reflexivity, check (a-a). Step 2: (a-a=0), and (0) is even. Step 3: If the condition is true for every element with itself, the relation is reflexive.

Correct answer: B. No

Step 1: For reflexivity, check (a+a). Step 2: (a+a=2a) is always even, not odd. Step 3: If ((a,a)) does not satisfy the condition, the relation is not reflexive.

Correct answer: A. Yes

A relation on A is reflexive when (a, a) belongs to R for every a in A. Here, 1 divides 1, 2 divides 2 and 3 divides 3, because every non-zero integer divides itself. Thus (1,1), (2,2) and (3,3) are all in R, so option A is correct. Options C and D check only part of the required condition.

Correct answer: B. No

Step 1: Check ((1,1),(2,2),(3,3)) for reflexivity. Step 2: (1+1=2) and (3+3=6), so not all self-pairs satisfy the condition. Step 3: Missing even one required self-pair means the relation is not reflexive.

Correct answer: A. Reflexive

Step 1: (|a-b|=0) holds exactly when (a=b). Step 2: So ((1,1),(2,2),(3,3)) are all included. Step 3: Equality-based relations are good examples of reflexivity.

Correct answer: A. ({(p,p),(q,q),(r,r)})

Step 1: A reflexive relation relates every element to itself. Step 2: For (p,q,r), the pairs ((p,p),(q,q),(r,r)) are required. Step 3: Even if symbols change, the rule stays the same.

Correct answer: B. The relation is not reflexive

Step 1: Since (4\in A), reflexivity requires ((4,4)). Step 2: This pair is missing, so the condition fails. Step 3: In checking reflexivity, any missing diagonal pair gives the answer quickly.

Correct answer: A. Yes

Step 1: Reflexivity requires only all pairs of the form ((a,a)). Step 2: Other types of pairs are not compulsory. Step 3: Do not confuse missing extra pairs with missing reflexive pairs.

Correct answer: A. (R) is reflexive

Step 1: First check the diagonal pairs. Step 2: ((1,1),(2,2),(3,3)) are all present. Step 3: Whatever the other pairs are, these three make the relation reflexive.

Correct answer: C. Every ((a,b)) must be present

Step 1: A reflexive relation does not require all possible pairs. Step 2: It only requires every ((a,a)) pair. Step 3: A relation with all pairs is universal, but reflexivity itself needs only self-pairs.

Correct answer: B. (5)

Step 1: The identity relation contains only ((a,a)) for each element. Step 2: The set (A) has five elements, so there are five pairs. Step 3: The number of pairs in the identity relation equals the number of elements.

Correct answer: A. Reflexive

Step 1: For (A={1,2}), ((1,1)) and ((2,2)) are required. Step 2: Both pairs are present. Step 3: This relation is also (A\times A), so it is reflexive.

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

Step 1: The only necessary reflexive pairs are of the form ((a,a)). Step 2: For (1,2,3), these are ((1,1),(2,2),(3,3)). Step 3: If the question says only necessary, do not choose extra pairs.

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

Step 1: The condition of reflexivity applies to every element. Step 2: So if (a\in A), then ((a,a)\in R) must hold. Step 3: In definition questions, connect the words with the symbolic form.

Correct answer: A. Yes

Step 1: For (0,1,2), the required pairs are ((0,0),(1,1),(2,2)). Step 2: All three are present in the relation. Step 3: The extra pair ((0,1)) does not affect reflexivity.

Correct answer: A. ((0,0))

Step 1: (0) is also an element of the set. Step 2: Reflexivity requires ((0,0)). Step 3: Add the missing self-pair to make the relation reflexive.

Correct answer: A. Identity relation

Step 1: Pairs only of the form ((a,a)) make the identity relation. Step 2: Here every element has its self-pair. Step 3: The identity relation is reflexive, but every reflexive relation need not be only identity.

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

Step 1: The identity relation contains only self-pairs. Step 2: The first option gives exactly those pairs. Step 3: If an extra pair is added, the relation may remain reflexive but is no longer identity.

Correct answer: A. At least one ((a,a)) pair is missing

Step 1: The main condition of reflexivity is the presence of all ((a,a)) pairs. Step 2: If even one is missing, the condition fails. Step 3: To identify a non-reflexive relation, look for a missing diagonal pair.

Correct answer: A. Yes

Step 1: For reflexivity, checking (a=a) is enough. Step 2: The condition includes (a=b), so every ((a,a)) is included. Step 3: If equality is included in the condition, check reflexivity immediately.

Correct answer: B. No

Step 1: Reflexivity needs pairs where (a=a). Step 2: But the condition (a\ne b) excludes all self-pairs. Step 3: Relations based on inequality are usually not reflexive.

Correct answer: A. Reflexive relation

Step 1: Every element being related to itself is the sign of reflexivity. Step 2: Symbolically, ((a,a)\in R) for every (a\in A). Step 3: Remember the definition by connecting it with simple classroom language.