Symmetric relation

Step 1: For symmetry, ((y,x)) must satisfy the same condition whenever ((x,y)) does. Step 2: If (x-y=2), then generally (y-x=2) is not true. Step 3: Be careful with subtraction-based conditions because order matters.

Step 1: In a symmetric relation, swapping order should not change the condition. Step 2: (a+b) and (b+a) are equal, so the condition of being even holds both ways. Step 3: Addition-based conditions are not affected by order.

Step 1: Look at the condition that (a+b) is even. Step 2: If (a+b) is even, then (b+a) is also even because changing order does not change the sum. Step 3: In sum-based conditions, reverse pairs are easy to verify.

Step 1: ((2,6)) and ((6,2)) are reverses of each other. Step 2: Such a pair supports the symmetry condition. Step 3: Remember, to call the whole relation symmetric, every pair must be checked.

Step 1: Find the reverses of pairs with different components. Step 2: ((2,1)) exists for ((1,2)), and ((4,2)) exists for ((2,4)). Step 3: ((3,3)) is automatically safe, so the relation is symmetric.

Step 1: First form the reverse of each given pair. Step 2: ((1,2)) and ((2,1)) already form a pair, but ((4,3)) is missing for ((3,4)). Step 3: Identify the missing part through the reverse pair.

Step 1: In a symmetric relation, the reversed pair must also be in the relation. Step 2: The reverse of ((8,3)) is ((3,8)). Step 3: Diagonal pairs are not required by this question.

Step 1: Check the reverse of every non-diagonal pair. Step 2: In the first option, both ((5,6)) and ((6,5)) are present, while ((7,7)) reverses to itself. Step 3: To decide quickly, look for lone off-diagonal pairs.

Step 1: Same parity means both are even or both are odd. Step 2: If (a) and (b) have the same parity, then (b) and (a) also have the same parity. Step 3: Such two-way conditions show symmetry.

Step 1: For symmetry, the relation must be true in both directions. Step 2: If (a) is brother of (b), (b) may be a sister, so (b) need not be brother of (a). Step 3: In real-life examples, read the meaning of the relation carefully.

Step 1: Being a classmate has the same meaning in both directions. Step 2: If (a) is a classmate of (b), then (b) is also a classmate of (a). Step 3: Connect two-way social relations with symmetry.

Step 1: (|a-b|) and (|b-a|) are equal. Step 2: So if (|a-b|=3), then (|b-a|=3) also holds. Step 3: Distance-like conditions using absolute value are often symmetric.

Step 1: For symmetry, the condition should remain true after swapping (a) and (b). Step 2: From (a^2=b^2), (b^2=a^2) is also true. Step 3: If equality remains true after swapping sides, symmetry holds.

Step 1: The reverse of ((1,2)) is ((2,1)). Step 2: The reverse of ((2,3)) is ((3,2)), and ((1,1)) is fine by itself. Step 3: When all reverse checks pass, the relation is symmetric.

Step 1: Symmetry requires the presence of the reverse order. Step 2: Therefore, if ((a,b)) is present, ((b,a)) must also be in the relation. Step 3: Remember this rule as the core definition.

Step 1: Reversing an ordered pair swaps its two positions. Step 2: In ((6,6)), both components are equal, so the reverse is the same pair. Step 3: Pairs of the form ((a,a)) are always safe for symmetry.

Step 1: Separate the pairs whose reverses already exist. Step 2: ((1,4)) and ((4,1)) are complete, and only ((5,2)) is needed for ((2,5)). Step 3: The number of missing reverse pairs gives the minimum number to add.

Step 1: ((2,1)) is needed for ((1,2)). Step 2: ((4,3)) is needed for ((3,4)). Step 3: Two different reverse pairs are missing, so two pairs must be added.

Step 1: Symmetry guarantees only the reverse pair. Step 2: From ((a,b)), ((b,a)) must follow, but ((a,a)) need not. Step 3: Do not mix the reflexive condition with the symmetric condition.

Step 1: First check symmetry: both ((1,2)) and ((2,1)) are present. Step 2: For reflexivity, ((1,1)) and ((2,2)) are also needed, which are absent here. Step 3: A symmetric relation need not be reflexive.