Symmetric relation

Correct answer: A. Symmetric

Step 1: When (a=b), the pair is of the form ((a,a)). Step 2: The reverse of ((a,a)) is the same pair. Step 3: A relation formed by equality is symmetric.

Correct answer: A. (R) is symmetric

Step 1: The reverse of ((1,2)), ((2,1)), is present. Step 2: ((1,1)) is its own reverse. Step 3: Symmetry does not require all diagonal pairs to be present.

Correct answer: A. Every ((a,a)) must be present for symmetry

Step 1: Symmetry asks only for reverses of the pairs that are present. Step 2: It does not require every ((a,a)) to be present. Step 3: Remember the difference between symmetric and reflexive relations.

Correct answer: A. No, reverse pairs must be checked separately

Step 1: Reflexivity talks about pairs of the form ((a,a)). Step 2: Symmetry talks about ((a,b)) and ((b,a)). Step 3: Do not mix different properties while deciding.

Correct answer: A. ((2,1))

Step 1: ((1,1)) and ((2,2)) are fine because they are their own reverses. Step 2: The reverse of ((1,2)), ((2,1)), is missing. Step 3: Even one missing reverse pair breaks symmetry.

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

Step 1: In option A, ((1,2)) is present. Step 2: Its reverse ((2,1)) is absent, so it is not symmetric. Step 3: In options, check non-diagonal pairs first.

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

Step 1: Symmetry checks the ordered pair after swapping its two positions. Step 2: So the correct form is ((a,b)\in R\Rightarrow (b,a)\in R). Step 3: Learn the meaning of the symbols, not only the statement.

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

Step 1: A pair is its own reverse when both entries are the same. Step 2: Reversing ((4,4)) gives ((4,4)). Step 3: Recognize such pairs as diagonal pairs.

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

Step 1: Symmetry only tells us that ((b,a)) is present. Step 2: It does not require ((a,a)) to be present. Step 3: Do not treat conclusions as compulsory unless they follow from the definition.

Correct answer: A. Symmetric

Step 1: If (a+b=5), then (b+a=5) also holds. Step 2: So the reverse pair ((b,a)) also belongs to the relation. Step 3: In sum-based rules, changing order does not change the rule.

Correct answer: A. Because if (a<b), then (b<a) is not true

Step 1: (a<b) is a one-direction rule. Step 2: ((1,2)) belongs, but ((2,1)) does not satisfy the rule. Step 3: If at least one reverse pair fails, the relation is not symmetric.

Correct answer: A. Not symmetric

Step 1: ((1,2)) belongs to the relation because (1\le 2). Step 2: ((2,1)) does not belong because (2\le 1) is false. Step 3: Order-based rules are often not symmetric.

Correct answer: A. Symmetric

Step 1: ((1,2)) has ((2,1)) with it. Step 2: ((2,3)) has ((3,2)), and ((1,1)) is safe. Step 3: Symmetry is proved when all reverse pairs are present.

Correct answer: A. Symmetric

Step 1: A symmetric relation must contain the reverse of every pair. Step 2: Here ((5,8)) is present but ((8,5)) is absent. Step 3: Even one missing reverse pair destroys symmetry.

Correct answer: A. ((a,b)) when (a+b) is even

Step 1: The value of (a+b) remains the same after changing order. Step 2: Therefore, if (a+b) is even, then (b+a) is also even. Step 3: When a rule is not affected by order, check for symmetry.

Correct answer: A. It is symmetric

Step 1: Both ((1,2)) and ((2,1)) are present. Step 2: Both ((2,3)) and ((3,2)) are present, and diagonal pairs are fine. Step 3: If every pair has its reverse, the relation is symmetric.

Correct answer: A. If one is present, the other will also be present

Step 1: Symmetry keeps a pair and its reverse together. Step 2: So if ((a,b)) is present, ((b,a)) must also be present. Step 3: Remember it as pairwise checking.

Correct answer: A. Yes, because both are reverses of each other

Step 1: The reverse of ((1,2)) is ((2,1)). Step 2: Both pairs are present, so symmetry is satisfied. Step 3: Diagonal pairs are not compulsory for symmetry.

Correct answer: A. When the reverse of a pair is absent

Step 1: The main condition of symmetry is the presence of reverse pairs. Step 2: If any reverse pair is missing, the condition fails. Step 3: One counterexample is enough to disprove symmetry.

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

Step 1: For a pair to be its own reverse, both entries must be equal. Step 2: In ((2,2)), both entries are equal. Step 3: Identifying diagonal pairs quickly helps in checking symmetry.

Correct answer: A. Symmetric

Step 1: If (|a-b|=1), then (|b-a|=1) also holds. Step 2: Therefore, the reverse pair also belongs to the relation. Step 3: In absolute difference rules, changing order does not change the value.

Correct answer: A. (R) is symmetric

Step 1: The reverse of ((1,3)), ((3,1)), is present. Step 2: ((2,2)) and ((3,3)) are their own reverses. Step 3: Only pairs present in the relation need their reverses.

Correct answer: A. Symmetric relation

Step 1: The given statement says the reverse pair is always present. Step 2: This is exactly the definition of a symmetric relation. Step 3: In name-identification questions, match the statement directly with the definition.

Correct answer: A. No, because its reverse is the same

Step 1: In ((6,6)), both entries are equal. Step 2: Reversing it still gives ((6,6)). Step 3: Treat pairs with equal entries as immediately safe for symmetry.

Correct answer: A. ((4,3))

Step 1: ((1,2)) and ((2,1)) are already reverse pairs. Step 2: The reverse of ((3,4)), ((4,3)), is missing. Step 3: Add the missing reverse pair to complete symmetry.