Symmetric relation

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.

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.

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.

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.

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.

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.

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.