Symmetric relation

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.

Step 1: A symmetric relation is identified by reverse pairs. Step 2: If ((a,b)) is present, ((b,a)) must also be present. Step 3: Remember a symmetric relation as a two-way pairing.

Step 1: In a symmetric relation, reversing an ordered pair must keep it in the relation. Step 2: So if ((a,b)\in R), then ((b,a)\in R) must also belong to (R). Step 3: In exams, always check the reverse of each pair.

Step 1: The reverse of ((1,2)), which is ((2,1)), is present. Step 2: The reverse of ((2,1)) is also present, so the relation is symmetric. Step 3: For small sets, list every pair and its reverse.

Step 1: Symmetry requires the reverse of every ordered pair. Step 2: Here ((1,2)) is present but ((2,1)) is missing. Step 3: Do not check only diagonal pairs; focus on reverse pairs.

Step 1: Reversing ((a,a)) gives the same pair ((a,a)). Step 2: Therefore diagonal pairs do not break symmetry. Step 3: In exams, pairs of the form ((a,a)) are always safe for symmetry.

Step 1: A symmetric relation must contain the reverse of every ordered pair. Step 2: The reverse of ((3,5)) is ((5,3)). Step 3: In such questions, swap the two positions first.

Step 1: ((1,1)) and ((2,2)) are their own reverses. Step 2: ((1,3)) has its reverse ((3,1)) present. Step 3: If every non-diagonal pair has its reverse, the relation is symmetric.

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

Step 1: If (a+b) is even, then (b+a) is also even. Step 2: So whenever ((a,b)) belongs to (R), ((b,a)) also belongs to (R). Step 3: In addition-based rules, changing order does not change the sum.

Step 1: If (a-b) is even, then (b-a) is also even. Step 2: Thus the reverse pair also satisfies the same rule. Step 3: In subtraction rules, the sign changes but evenness does not.

Step 1: Diagonal pairs are their own reverses. Step 2: Both ((1,2)) and ((2,1)) are present. Step 3: When explaining, connect the answer directly with the definition.