Symmetric relation

Answer: B, 32. In a symmetric relation, whenever (a,b) is present, (b,a) must also be present. Since (1,2) is required, (2,1) is forced as well; these two pairs together represent one fixed off-diagonal choice. The remaining independent choices are the three diagonal pairs (1,1), (2,2), (3,3), and the two off-diagonal pair-groups represented by (1,3),(3,1) and (2,3),(3,2). Thus there are 5 independent binary choices, giving 2^5=32 relations. Option A counts only four choices, C counts six, and D counts seven; each incorrectly treats the forced reverse pair or another linked pair as independent. Memory cue: under symmetry, count unordered pairs, not separate directions.

Step 1: The reverse of ((1,2)) is ((2,1)), and the reverse of ((2,4)) is ((4,2)). Step 2: ((3,3)) is its own reverse, so it does not break symmetry. Step 3: In exams, check non-diagonal pairs in reverse pairs.

Step 1: ((1,3)) and ((3,1)) are already reverses of each other. Step 2: The reverse of ((2,4)), which is ((4,2)), is missing. Step 3: To make a relation symmetric, add only the missing reverse pair.

Step 1: If (a+b) is even, then (b+a) is also even. Step 2: Hence if ((a,b)\in R), then ((b,a)\in R) too. Step 3: In addition-based rules, changing order does not change the sum, so symmetry is easy to test.

Step 1: If (a-b) is odd, then (b-a) is also odd because only the sign changes. Step 2: Therefore the reverse pair also satisfies the same rule. Step 3: While checking evenness or oddness, do not worry about the negative sign.

Step 1: In (R^{-1}), every ordered pair is reversed. Step 2: If (R^{-1}=R), then every reverse pair already belongs to (R). Step 3: Treat equality with the inverse as a quick test for symmetry.

Step 1: First check the reverses of the given non-diagonal pairs. Step 2: ((3,4)) is present, but its reverse ((4,3)) is missing. Step 3: One missing reverse pair is enough to break symmetry.

Step 1: (1\le 2) is true, so ((1,2)\in R). Step 2: (2\le 1) is false, so ((2,1)\notin R). Step 3: In order-based rules, always check the reverse pair separately.

Step 1: In absolute difference, (|a-b|) and (|b-a|) are equal. Step 2: So if (|a-b|=2), then (|b-a|=2) also holds. Step 3: Absolute value rules are often good examples of symmetry.

Step 1: Symmetry guarantees only the reverse pair. Step 2: The reverse of ((5,7)) is ((7,5)), so it must be present. Step 3: Symmetry does not automatically prove all diagonal pairs.

Step 1: ((1,2)) has ((2,1)), and ((2,3)) has ((3,2)). Step 2: ((1,1)) and ((2,2)) are their own reverses. Step 3: Even if ((3,3)) is missing, symmetry may still hold; that is a reflexivity issue.

Step 1: In option A, both ((1,2)) and ((2,1)) are present, so it is symmetric. Step 2: It does not contain all ((1,1),(2,2),(3,3)), so it is not reflexive. Step 3: In questions with two properties, check both conditions separately.

Step 1: Option A has diagonal pairs ((1,1)) and ((2,2)). Step 2: But the reverse of ((1,3)), which is ((3,1)), is missing, so it is not symmetric. Step 3: Having diagonal pairs does not automatically prove symmetry.

Step 1: Match the entries on both sides of the main diagonal. Step 2: Here (m_{13}=m_{31}=1), (m_{12}=m_{21}=0), and (m_{23}=m_{32}=0). Step 3: A matrix identical about the main diagonal represents a symmetric relation.

Step 1: (m_{23}=1) means ((2,3)\in R). Step 2: (m_{32}=0) means ((3,2)\notin R). Step 3: If the reverse of one pair is missing, the relation is not symmetric.

Step 1: (A\times A) contains all possible ordered pairs from (A). Step 2: Therefore the reverse ((b,a)) of any ((a,b)) is also present. Step 3: Remember that the universal relation is an important example of a symmetric relation.

Step 1: The condition of symmetry is checked only for pairs that are present. Step 2: The empty relation has no pair, so there is no counterexample. Step 3: The empty relation is treated as symmetric automatically.

Step 1: (a\equiv b \pmod{2}) means (a) and (b) have the same parity. Step 2: If (a) has the same parity as (b), then (b) has the same parity as (a). Step 3: In similarity-type rules, check both directions naturally.

Step 1: If (a\equiv b \pmod{3}), both leave the same remainder when divided by (3). Step 2: The same fact also gives (b\equiv a \pmod{3}). Step 3: Congruence rules are good examples of symmetry.

Step 1: (2=1+1), so ((2,1)\in R). Step 2: For ((1,2)), (1=2+1) is false. Step 3: One-direction rules often fail symmetry.

Step 1: If (a^2=b^2), then by reversing equality, (b^2=a^2) is also true. Step 2: Hence if ((a,b)) belongs to the relation, ((b,a)) also belongs. Step 3: In equality-based rules, swap the two sides to test symmetry.

Step 1: The pairs giving sum (5) are ((1,4),(2,3),(3,2),(4,1)). Step 2: The reverse of every pair is also in the list, so the relation is symmetric. Step 3: In counting questions, first list all pairs systematically.

Step 1: ((3,1)\in R) because (3-1=2). Step 2: ((1,3)\notin R) because (1-3=-2), not (2). Step 3: In fixed difference rules, reversing may change the sign, so be careful.

Step 1: (|a-b|) and (|b-a|) always have the same value. Step 2: Therefore, if (|a-b|\le 1), the reverse pair also satisfies the condition. Step 3: Absolute difference inequalities often lead to symmetric relations.

Step 1: In a symmetric relation, the reverse of every given pair must also exist. Step 2: The reverse of ((2,5)) is ((5,2)), and the reverse of ((4,6)) is ((6,4)). Step 3: For multiple pairs, reverse each pair separately.