Symmetric relation

Step 1: For symmetry, the condition should remain valid after swapping (a) and (b). Step 2: In (a^2+b^2=10), swapping gives (b^2+a^2=10), which is the same condition. Step 3: Balanced sum-type conditions often produce symmetric relations.

Step 1: If (a^2=b^2), then the reversed equality (b^2=a^2) is also true. Step 2: Hence ((a,b)\in R) implies ((b,a)\in R). Step 3: Equality-based relations are symmetric when swapping the two sides keeps the statement true.

Step 1: Absolute difference satisfies (|a-b|=|b-a|). Step 2: If (|a-b|\leq2), then (|b-a|\leq2) also holds. Step 3: Distance-based bounds often produce symmetric relations.

Step 1: If (a+b\equiv 0 \pmod{3}), then (b+a\equiv 0 \pmod{3}) also holds. Step 2: Changing the order of addition does not change the remainder. Step 3: Modular conditions based on sums are usually symmetric.

Step 1: (a-b\equiv 0 \pmod{3}) means (a) and (b) have the same remainder modulo (3). Step 2: Then (b-a\equiv 0 \pmod{3}) also holds. Step 3: Congruence equality conditions remain valid after reversing the pair.

Step 1: Symmetry requires the reverse pair to satisfy the same condition. Step 2: For ((2,1)), (2-1\equiv 1 \pmod{3}), but for ((1,2)), (1-2\equiv 2 \pmod{3}), so the condition fails. Step 3: Non-zero modular difference conditions must be checked carefully.

Step 1: Check the reverse of every off-diagonal pair. Step 2: ((1,2)) has ((2,1)), and ((3,1)) has ((1,3)). The pair ((1,1)) is its own reverse. Step 3: ((2,2)) and ((3,3)) are not compulsory for symmetry.

Step 1: The definition of symmetry is directly about the reverse ordered pair. Step 2: If ((a,b)\in R), then ((b,a)\in R) must be true. Step 3: Diagonal pairs are related to reflexivity, not necessarily to symmetry.

Step 1: (a=b+1) is a directional condition. Step 2: ((2,1)) satisfies it, but ((1,2)) does not. The other conditions remain the same after swapping (a) and (b). Step 3: Be cautious when the roles of (a) and (b) are not balanced.

Step 1: The condition says both numbers are prime; it does not depend on direction. Step 2: If (a) and (b) are both prime, then (b) and (a) are also both prime. Step 3: Conditions where both variables have the same role usually preserve symmetry.

Step 1: Having the same parity is a two-way condition. Step 2: If (a) and (b) are both even or both odd, then (b) and (a) are also the same way. Step 3: Conditions based on belonging to the same class give symmetric relations.

Step 1: An arrow from (a) to (b) represents ((a,b)). Step 2: A reverse arrow for every arrow means ((b,a)) is also present, which is symmetry. Step 3: In diagram-based questions, observe arrow directions carefully.

Step 1: ((1,2)) and ((2,1)) are already balanced. Step 2: ((3,1)) is present, but its reverse ((1,3)) is missing. ((2,2)) is its own reverse. Step 3: Add only missing reverse pairs, not unnecessary diagonal pairs.

Step 1: (ab) and (ba) are equal. Step 2: If (ab) is even, then (ba) is also even, so the reverse pair also belongs to the relation. Step 3: Multiplication is commutative, so product-based conditions often preserve symmetry.

Step 1: Swapping (a) and (b) in (a+b=ab) gives (b+a=ba). Step 2: Both addition and multiplication are commutative, so the condition remains the same. Step 3: If the full condition is unchanged after swapping, the relation is symmetric.

Step 1: (a^2-b^2=0) means (a^2=b^2). Step 2: If (a^2=b^2), then (b^2=a^2) is also true, so the reverse pair also belongs to the relation. Step 3: Simplifying the condition helps in exam questions.

Step 1: The difference being divisible by (2) means (a-b) is even. Step 2: If (a-b) is even, then (b-a) is also even. Step 3: Difference conditions with zero remainder remain valid after reversing the pair.

Step 1: ((1,2)) and ((2,1)) are both present, so they are fine. Step 2: ((2,3)) and ((3,2)) are also both present. Only the reverse of ((1,3)), which is ((3,1)), is missing. Step 3: Check off-diagonal pairs in reverse-pair groups.

Step 1: If (a+b) is a multiple of (5), then (b+a) gives the same sum. Step 2: Hence ((b,a)) belongs to the relation whenever ((a,b)) does. Step 3: For sum-multiple conditions, whether the modulus is even or odd does not affect symmetry.

Step 1: In (R^{-1}), all ordered pairs are reversed. Step 2: (R=R^{-1}) exactly when every reverse pair is already in (R), which is symmetry. Step 3: (R=R^{-1}) is a very useful test for symmetry.

Step 1: The reverse of ((3,4)) is ((4,3)). Step 2: (3+4=7) and (4+3=7), so the reverse pair also satisfies the relation. Step 3: In sum conditions, reverse pairs can be checked quickly.

Step 1: For a relation matrix to represent a symmetric relation, entries across the main diagonal must match. Step 2: Here (m_{13}=1) and (m_{31}=1), and the other opposite entries also match. Step 3: A relation is symmetric when its matrix equals its transpose.

Step 1: For symmetry in a matrix, we need (m_{ij}=m_{ji}). Step 2: Here (m_{12}=1), but (m_{21}=0), so a reverse pair is missing. Step 3: In matrix questions, compare entries across the main diagonal.

Step 1: The reverse of a diagonal pair ((a,a)) is the same pair ((a,a)). Step 2: So the symmetry condition is automatically satisfied for every such pair. Step 3: Any relation containing only diagonal pairs is symmetric.

Step 1: From (a+b=2a), we get (b=a). Step 2: So the relation contains only diagonal pairs ((a,a)), and each is its own reverse. Step 3: Simplify the condition first, then check symmetry.