Symmetric relation

Step 1: For ((1,2)), the reverse pair ((2,1)) is required. Step 2: Since it is missing, the definition of symmetry fails. Step 3: One counterexample is enough to disprove symmetry.

Step 1: In a symmetric relation, the presence of ((i,j)) and ((j,i)) must match. Step 2: In matrix form, this means (m_{ij}=m_{ji}). Step 3: This condition is written as (M=M^T).

Step 1: In the given (R), every pair has its reverse, so it is symmetric. Step 2: ((1,2)) and ((2,3)) are present, but ((1,3)) is missing, so transitivity fails. Step 3: Symmetry does not automatically imply transitivity.

Step 1: Symmetry fails when a present pair lacks its reverse. Step 2: If both ((2,3)) and ((3,2)) are absent, there is no imbalance. Step 3: Absence in both directions does not break symmetry.

Step 1: Reversing a diagonal pair ((a,a)) gives the same pair ((a,a)). Step 2: So the reverse of each such pair is already present. Step 3: Diagonal pairs are always safe for symmetry.

Step 1: If (a) and (b) are both even or both odd, the same remains true after swapping them. Step 2: Therefore ((a,b)) implies ((b,a)). Step 3: Same-type conditions often give symmetric relations.

Step 1: If (a^2=b^2), then reversing the equality gives (b^2=a^2). Step 2: So ((a,b)) implies ((b,a)). Step 3: Equality-based conditions are easy to test by swapping sides.

Step 1: Symmetry means the relation works in both directions. Step 2: For friendship to be symmetric, if one person is a friend of another, the second must also be a friend of the first. Step 3: Even in real-life examples, check the reverse direction.

Step 1: Both ((1,2)) and ((2,1)) are present, so symmetry holds. Step 2: ((1,1),(2,2),(3,3)) are all present, so reflexivity also holds. Step 3: When two properties are asked, verify both separately.

Step 1: Symmetry depends on every pair and its reverse pair. Step 2: Therefore, all non-diagonal pairs must be checked in the reverse direction. Step 3: Do not conclude by checking only one pair in a hurry.

Step 1: In a symmetric relation, if ((a,b)) is present, then ((b,a)) must also be present. Step 2: ((2,3)) is present but ((3,2)) is missing. The pair ((1,2)) already has ((2,1)). Step 3: In exams, always check the reverse of every non-diagonal pair.

Step 1: If (a+b) is even, then (b+a) is also even because changing the order of addition does not change the sum. Step 2: So whenever ((a,b)\in R), we also get ((b,a)\in R). Step 3: If the condition remains unchanged after swapping (a) and (b), symmetry is usually satisfied.

Step 1: (a-b) being even means (a) and (b) have the same parity. Step 2: Then (b-a) is also even, so the reverse pair also belongs to the relation. Step 3: For difference-based conditions, the sign may change, but parity remains the same.

Step 1: Symmetry only requires that every ((a,b)) has ((b,a)). Step 2: Here ((1,2)) has ((2,1)), and ((2,3)) has ((3,2)). A diagonal pair is its own reverse. Step 3: All diagonal pairs are not required for symmetry.

Step 1: In a symmetric relation, reversing an ordered pair must still give a pair in the relation. Step 2: The reverse of ((4,7)) is ((7,4)), so it must be present. Step 3: Symmetry does not automatically force diagonal pairs like ((4,4)) or ((7,7)).

Step 1: One counterexample is enough to disprove symmetry. Step 2: ((1,2)\in R) because (1\leq2), but ((2,1)\notin R) because (2\leq1) is false. Step 3: Order-based relations often fail to be symmetric.

Step 1: (|a-b|=|b-a|) is always true. Step 2: So if ((a,b)) satisfies the relation, ((b,a)) also satisfies the same condition. Step 3: Conditions based on absolute distance are usually symmetric.

Step 1: Symmetry requires ((b,a)) whenever ((a,b)) is present. Step 2: ((1,2)\in R) because (1) divides (2), but ((2,1)\notin R). Step 3: In divisibility relations, direction matters.

Step 1: If (a+b=6), then (b+a=6) also holds. Step 2: Thus the reverse ((b,a)) belongs to the relation whenever ((a,b)) does. Step 3: Sum-based equality conditions usually remain unchanged after swapping the variables.

Step 1: In a symmetric relation, the reverse of each given pair is compulsory. Step 2: From ((1,3)), we need ((3,1)), and from ((2,3)), we need ((3,2)). Step 3: Only the required reverse pairs are guaranteed, not unrelated diagonal or other pairs.