Symmetric relation

Step 1: A symmetric example must contain a pair along with its reverse. Step 2: Option A has both ((1,2)) and ((2,1)). Step 3: Even in small examples, do not forget to match reverse pairs.

Step 1: In a symmetric relation, the reverse of every ordered pair must also be in the relation. Step 2: ((1,4)) is matched by ((4,1)), and ((2,2)) reverses to itself. Step 3: In such questions, first make pairs of reverse ordered pairs.

Step 1: For symmetry, reverse the given ordered pair. Step 2: The reverse of ((10,3)) is ((3,10)). Step 3: Symmetry guarantees only the reverse pair, not all other related pairs.

Step 1: ((1,2)) and ((2,1)) are already reverse pairs. Step 2: The reverse of ((3,2)), which is ((2,3)), is missing. Step 3: To make a relation symmetric, add only the missing reverse pair.

Step 1: In a symmetric relation, every pair with different entries must have its reverse pair. Step 2: Option B has ((1,3)) with ((3,1)), and ((2,2)) reverses to itself. Step 3: To solve options quickly, look for missing reverse pairs.

Step 1: The reverse of ((1,1)) is itself. Step 2: ((1,2)) is matched by ((2,1)), and ((2,3)) is matched by ((3,2)). Step 3: Symmetry does not require all ((a,a)) pairs to be present.

Step 1: If (a-b) is even, then its negative (b-a) is also even. Step 2: So whenever ((a,b)) satisfies the rule, ((b,a)) also satisfies it. Step 3: For rule-based relations, check whether the reversed pair satisfies the same rule.

Step 1: If (|a-b|=1), then reversing the order gives (|b-a|=1) as well. Step 2: Therefore, if ((a,b)) is in the relation, ((b,a)) is also in it. Step 3: Rules involving absolute difference often produce symmetry.

Step 1: In option D, ((2,3)) is matched by ((3,2)). Step 2: But the reverse of ((1,2)), which is ((2,1)), is missing. Step 3: If even one pair is missing its reverse, the relation is not symmetric.

Step 1: Reversing ((5,5)) gives ((5,5)) again. Step 2: Reversing ((7,7)) gives ((7,7)) again. Step 3: Pairs with the same entries satisfy the symmetry condition.

Step 1: Symmetry is related to the order of entries in a pair. Step 2: So we must check whether ((b,a)) is present for every ((a,b)). Step 3: Do not decide only by counting the number of pairs.

Step 1: ((1,1)) and ((3,3)) are their own reverses. Step 2: ((2,3)) is matched by ((3,2)). Step 3: Since all pairs have their reverses, the relation is symmetric.

Step 1: The reverse of ((2,5)) is ((5,2)). Step 2: A symmetric relation must contain this reverse pair. Step 3: If the reverse pair is missing, the relation cannot be symmetric.

Step 1: The main idea in a symmetric relation is the reverse pair. Step 2: So if ((a,b)) is present, ((b,a)) must be present. Step 3: In exams, do not fear symbolic notation; just reverse the order of the pair.

Step 1: The pair ((1,2)) is already matched by ((2,1)). Step 2: The reverse of ((3,4)), which is ((4,3)), is missing. Step 3: Adding the reverse of the incomplete pair will make the relation symmetric.

Step 1: Same parity means both numbers are even or both are odd. Step 2: If (x) and (y) have the same parity, then (y) and (x) also have the same parity. Step 3: Rules that remain true in both directions give symmetry.

Step 1: A pair is equal to its own reverse when both entries are the same. Step 2: Among the given pairs, reversing ((2,2)) gives ((2,2)) again. Step 3: Pairs like ((a,a)) are always safe in symmetry checks.

Step 1: ((1,2)) and ((2,1)) form a complete reverse pair. Step 2: ((2,3)) and ((3,2)) also form a complete reverse pair, but ((3,1)) is missing for ((1,3)). Step 3: Missing ((a,a)) pairs do not automatically break symmetry.

Step 1: In a symmetric relation, the reverse direction is included with the original direction. Step 2: So if ((m,n)) is present, ((n,m)) must also be present. Step 3: Remember it as a two-way pair.

Step 1: Pairs with (a=b) are their own reverses. Step 2: If (a+b=4), then (b+a=4) also holds, so the reverse pair satisfies the rule. Step 3: When the rule remains true after reversing the order, symmetry follows.

Step 1: ((1,4)) and ((4,1)) already form a complete reverse pair. Step 2: The reverse of ((2,3)), which is ((3,2)), is missing. Step 3: Therefore, adding one pair is enough.

The governing concept is symmetry, not reflexivity. A relation R on A is symmetric if, whenever (a,b) belongs to R, the reverse pair (b,a) also belongs to R. The empty relation satisfies this condition vacuously because it has no pair that could violate it. The universal relation A x A contains both directions of every pair, so it is symmetric. A self-pair (a,a) is its own reverse and therefore never creates a symmetry violation. However, symmetry does not require self-pairs for every element; that additional requirement defines reflexivity. Therefore option D is the false statement.

Step 1: A pair is equal to its own reverse when both entries are the same. Step 2: Here ((3,3)) and ((4,4)) are such pairs. Step 3: Do not count ((a,b)) and ((b,a)) as self-reverse when (a \neq b).

Step 1: (A \times A) contains all possible ordered pairs from (A). Step 2: So along with ((1,2)), its reverse ((2,1)) will also be present. Step 3: In the universal relation, reverse pairs do not need to be added separately.

Step 1: ((1,2)) and ((2,1)) are reverse pairs. Step 2: ((1,1)) is equal to its own reverse. Step 3: Symmetry needs reverses of the pairs that are present, not all possible pairs.