Symmetric relation

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.

Step 1: When (a=b), the pair is of the form ((a,a)). Step 2: The reverse of ((a,a)) is the same pair. Step 3: A relation formed by equality is symmetric.

Step 1: The reverse of ((1,2)), ((2,1)), is present. Step 2: ((1,1)) is its own reverse. Step 3: Symmetry does not require all diagonal pairs to be present.

Step 1: Symmetry asks only for reverses of the pairs that are present. Step 2: It does not require every ((a,a)) to be present. Step 3: Remember the difference between symmetric and reflexive relations.

Step 1: Reflexivity talks about pairs of the form ((a,a)). Step 2: Symmetry talks about ((a,b)) and ((b,a)). Step 3: Do not mix different properties while deciding.

Step 1: ((1,1)) and ((2,2)) are fine because they are their own reverses. Step 2: The reverse of ((1,2)), ((2,1)), is missing. Step 3: Even one missing reverse pair breaks symmetry.

Step 1: In option A, ((1,2)) is present. Step 2: Its reverse ((2,1)) is absent, so it is not symmetric. Step 3: In options, check non-diagonal pairs first.