Symmetric relation

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.

Step 1: First identify the non-diagonal pairs. Step 2: In option A, both ((1,2)) and ((2,1)) are present, and ((3,3)) is self-reverse. Step 3: In options, quickly spot missing reverse pairs.

Step 1: Reversing ((4,4)) gives ((4,4)) itself. Step 2: So no additional reverse pair is needed. Step 3: Pairs with equal entries do not create a problem for symmetry.

Step 1: Symmetry guarantees only the reverse pair. Step 2: The reverse of ((7,2)) is ((2,7)), so that must be present. Step 3: Symmetry does not automatically prove every diagonal pair.

The governing test for symmetry concerns ordered pairs and their reverses. For a relation R on A, take any pair (a,b) in R and check whether (b,a) is also in R. This check must be applied to every listed pair; finding even one pair whose reverse is absent proves that the relation is not symmetric. The number of elements in the set does not decide symmetry, because sets of the same size can support both symmetric and non-symmetric relations. The largest element is irrelevant, and a relation’s name cannot replace verification from its pairs. Hence option A states the correct first step and the exact operational test.

Step 1: The reverse of ((1,2)), ((2,1)), is present. Step 2: The reverse of ((3,2)), ((2,3)), is also present. Step 3: All non-diagonal pairs occur in reverse pairs, so the relation is symmetric.

Step 1: ((1,3)) and ((3,1)) are already balanced. Step 2: The reverse of ((2,3)), which is ((3,2)), is missing. Step 3: To make a relation symmetric, unnecessary diagonal pairs are not required.

Step 1: The condition of symmetry applies to every pair that is present. Step 2: The empty relation has no pair, so the condition is not violated. Step 3: In an empty case, the condition is treated as automatically true.

Step 1: (A\times A) contains all possible ordered pairs from (A). Step 2: Hence if ((a,b)) is present, ((b,a)) is also present. Step 3: Remember that the universal relation is symmetric.

The inverse relation R⁻¹ is formed by reversing every ordered pair: (a, b) ∈ R⁻¹ exactly when (b, a) ∈ R. If R is symmetric, the presence of (b, a) in R guarantees the presence of (a, b) in R, so reversing all pairs produces no new relation; every pair of R⁻¹ is already in R. The same reasoning in reverse gives R ⊆ R⁻¹, hence R⁻¹ = R. Symmetry does not imply emptiness, so options B and C are false, while D states the opposite. Therefore A is correct.

Step 1: (R^{-1}=R) means every reverse pair is also in the same relation. Step 2: This is exactly the main condition of a symmetric relation. Step 3: When you see equality with the inverse relation, identify symmetry quickly.

Step 1: If entries on both sides of the main diagonal match, reverse pairs occur together. Step 2: This represents the condition of symmetry. Step 3: In matrix questions, treat the main diagonal like a mirror line.

Step 1: (m_{12}=1) means ((1,2)) belongs to the relation. Step 2: By symmetry, ((2,1)) also belongs, so (m_{21}=1). Step 3: In a matrix, reversing a pair reverses the indices.

Step 1: All pairs are of the form ((a,a)). Step 2: The reverse of each such pair is the same pair, so symmetry holds. Step 3: The identity relation is always symmetric.

Step 1: Symmetry simply means the relation remains true in both directions. Step 2: If ((a,b)) is present, ((b,a)) must also be present. Step 3: Remember it as a two-way relation.