Symmetric relation

Correct answer: A. (1)

Step 1: The pair ((1,2)) and ((2,1)) is complete. Step 2: The pair ((1,3)) and ((3,1)) is also complete; only the reverse ((4,2)) of ((2,4)) is missing. Step 3: For minimum addition, count only missing reverse pairs.

Correct answer: A. (R)

Step 1: In the inverse relation, all ordered pairs are reversed. Step 2: Here the reverse of every pair is already in the same relation. Step 3: When a relation is symmetric, its inverse is equal to itself.

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

Step 1: (R^{-1}=R) holds exactly when the relation is symmetric. Step 2: In option A, both ((1,2)) and ((2,1)) are present, and ((3,3)) is its own reverse. Step 3: Convert inverse questions into symmetry checks.

Correct answer: A. Symmetric

Step 1: The greatest common divisor of two numbers does not change when the order changes. Step 2: If (\gcd(a,b)=1), then (\gcd(b,a)=1) also. Step 3: In number-property rules, swap the order and see whether the rule remains unchanged.

Correct answer: A. (R) is symmetric

Step 1: If (a) and (b) are both even, then after swapping, (b) and (a) are also both even. Step 2: Therefore, with ((a,b)), the reverse ((b,a)) also belongs to the relation. Step 3: When the same condition applies to both entries, check symmetry carefully.

Correct answer: A. ((1,2)\in R) but ((2,1)\notin R)

Step 1: (1) divides (2), so ((1,2)\in R). Step 2: (2) does not divide (1), so ((2,1)\notin R). Step 3: Divisibility generally does not remain the same after reversing the order.

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

Step 1: In option A, every non-diagonal pair has its reverse. Step 2: It does not contain all diagonal pairs, yet symmetry holds. Step 3: Keep the conditions of symmetry and reflexivity separate.

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

Step 1: Transitivity gives ((a,c)) from ((a,b)) and ((b,c)). Step 2: Here ((1,2)) and ((2,1)) give ((1,1)). Step 3: In mixed-property questions, think by chaining ordered pairs.

Correct answer: A. ((b,a))

Step 1: When (a\ne b), the reverse of ((a,b)) is a different pair ((b,a)). Step 2: This reverse pair is required for symmetry. Step 3: For non-diagonal pairs, do not forget to write the reverse separately.

Correct answer: A. ((a,b)) when (a<b)

Step 1: (a<b) is a one-direction rule. Step 2: If (a<b), then generally (b<a) is not true. Step 3: Rules depending on order often break symmetry.

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

Step 1: Having every ((a,a)) is the condition of reflexivity, not symmetry. Step 2: Symmetry asks only for reverses of existing pairs. Step 3: Distinguish between always true and not necessarily true statements.

Correct answer: A. (R) is symmetric but not reflexive

Step 1: ((1,2)) and ((2,1)), as well as ((1,3)) and ((3,1)), are reverse pairs. Step 2: ((1,1),(2,2),(3,3)) are missing, so it is not reflexive. Step 3: Check symmetry first and reflexivity separately.

Correct answer: A. Reflexive and symmetric

Step 1: All diagonal pairs ((1,1),(2,2),(3,3)) of (A) are present, so the relation is reflexive. Step 2: Both ((1,2)) and ((2,1)) are present, so it is also symmetric. Step 3: When two properties are asked, verify both fully.

Correct answer: A. (\begin{pmatrix}1&1\1&0\end{pmatrix})

Step 1: In a symmetric matrix, (m_{12}=m_{21}) must hold. Step 2: In option A, both entries are (1), so the matrix is identical about the main diagonal. Step 3: In small matrices, compare entries above and below the diagonal.

Correct answer: A. No extra pair is required

Step 1: The reverse of ((3,3)) is ((3,3)) itself. Step 2: Therefore no new reverse pair is needed because of this pair. Step 3: Treat diagonal pairs as immediately safe for symmetry.

Correct answer: A. Symmetric

Step 1: If (a\ne b), then (b\ne a) is also true. Step 2: So with ((a,b)), the reverse ((b,a)) also belongs to the relation. Step 3: This inequality rule remains the same in both directions.

Correct answer: A. Symmetric

Step 1: In addition, (a+b=b+a). Step 2: Therefore, if (a+b\le 5), then (b+a\le 5) also holds. Step 3: In sum-based inequalities, changing order does not change the condition.

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

Step 1: (1+2=3), which is odd, so ((1,2)) belongs to the relation. Step 2: (2+1=3) is also odd, so ((2,1)) also belongs. Step 3: In sum-based rules, the reverse pair usually satisfies the same condition.

Correct answer: A. Based on non-diagonal pairs, it can be symmetric

Step 1: Every given non-diagonal pair has its reverse present. Step 2: Diagonal pairs may or may not be present; they are not compulsory for symmetry. Step 3: In symmetry, check reverses only for existing pairs.

Correct answer: A. ((a,b)\in R) when (|a-b|=4)

Step 1: The value of (|a-b|) does not change after swapping the order. Step 2: Therefore, if (|a-b|=4), then (|b-a|=4) too. Step 3: Treat absolute difference as a quick clue for symmetry.

Correct answer: A. A pair ((a,b)\in R) for which ((b,a)\notin R)

Step 1: One counterexample is enough to disprove symmetry. Step 2: You need a pair whose reverse is not in the relation. Step 3: In a long list, first search for one missing reverse pair.

Correct answer: A. ((3,2)) is absent

Step 1: All diagonal pairs are present, so they are not the problem. Step 2: The reverse of ((2,3)), which is ((3,2)), is not in the relation. Step 3: Even when reflexivity appears to hold, check symmetry separately.

Correct answer: A. Symmetric

Step 1: (a+b) and (b+a) are always equal. Step 2: If a pair gives sum (6), its reverse will also give sum (6). Step 3: Connect the commutative property of addition with symmetry.

Correct answer: A. Add ((2,1)) to (R={(1,2),(3,3)})

Step 1: ((3,3)) is its own reverse, so it is fine. Step 2: The reverse of ((1,2)), which is ((2,1)), is missing. Step 3: For minimum correction, add only the required reverse pair.

Correct answer: A. ((4,1))

Step 1: The pair ((1,2)) and ((2,1)) is complete. Step 2: The pair ((2,4)) and ((4,2)) is also complete, but the reverse ((4,1)) of ((1,4)) is missing. Step 3: In a symmetric extension, every missing reverse pair is added.