Symmetric relation

Step 1: ((1,1)) and ((2,2)) are their own reverses. Step 2: Both ((1,2)) and ((2,1)) are present. Step 3: Therefore, every pair has its reverse in the relation.

Step 1: The reverse of ((2,3)) is ((3,2)). Step 2: The first option contains this required pair. Step 3: Other diagonal pairs do not fill the missing reverse pair.

Step 1: Living in the same city is a mutual condition. Step 2: If (a) lives in the same city as (b), then (b) lives in the same city as (a). Step 3: Recognize mutual same-property relations as symmetric.

Step 1: Being greater than is a direction-based relation. Step 2: If (a>b), then (b>a) cannot be true. Step 3: Ordered comparison relations are usually not symmetric.

Step 1: The condition contains (a^2+b^2). Step 2: Swapping (a) and (b) gives (b^2+a^2=10), which is the same condition. Step 3: Addition keeps the condition symmetric after swapping.

Step 1: Symmetry is checked by reverse pairs, not just by counting pairs. Step 2: For every ((a,b)), ((b,a)) should be found. Step 3: In exams, mark pairs in the list and match their reverses.

Step 1: ((1,1)) and ((2,2)) do not change after reversal. Step 2: Symmetry requires reverses only for existing pairs, not all possible pairs. Step 3: Missing extra pairs do not automatically break symmetry.

Step 1: ((1,2)) and ((2,1)) form one reverse pair. Step 2: ((3,1)) and ((1,3)) form another reverse pair. Step 3: Identifying reverse pairs helps in checking symmetry.

Step 1: Lack of symmetry usually comes from a missing reverse pair. Step 2: If ((a,b)) is present but ((b,a)) is not, the relation is not symmetric. Step 3: Wrong options often reverse the correct definition.

Step 1: If (a-b) is even, then (b-a=-(a-b)) is also even. Step 2: Therefore, whenever ((a,b)) satisfies the condition, ((b,a)) also satisfies it. Step 3: In parity questions, remember that the negative of an even number is also even.

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.