Symmetric relation

The correct answer is A: R is symmetric. To test symmetry, reverse each pair. The reverse of (1,1) is itself, and the reverse of (1,2) is (2,1), which is present. Therefore every pair has its reverse. R is not reflexive because a reflexive relation on {1,2} must contain both (1,1) and (2,2), but (2,2) is missing. It is not antisymmetric because for distinct elements 1 and 2, both (1,2) and (2,1) are present. Since an equivalence relation must be reflexive, symmetric, and transitive, the failure of reflexivity already rules out option D. Option B overlooks the missing diagonal pair, and C contradicts the definition of antisymmetry. Memory cue: for a two-element set, check the two diagonals for reflexivity and the two opposite off-diagonal pairs for symmetry or antisymmetry.

Step 1: In a symmetric relation, if ((a,b) \in R), then ((b,a) \in R) must also be present. Step 2: Here ((1,2)) has ((2,1)), and ((3,3)) reverses to itself. Step 3: In exams, always check the reverse pair of each ordered pair.

Step 1: In a symmetric relation, every pair must have its reverse pair. Step 2: The reverse of ((4,7)) is ((7,4)). Step 3: In such questions, reverse the order only; do not change the numbers.

Step 1: The reverse of ((1,1)) is the same pair, so it is fine. Step 2: The reverse of ((1,2)), which is ((2,1)), is missing. Step 3: To make a relation symmetric, add the missing reverse pair.

Step 1: The main rule of a symmetric relation is about reversing an ordered pair. Step 2: If ((a,b)) is present, ((b,a)) must also be present. Step 3: Keep this rule separate from reflexive and transitive conditions.

Step 1: Symmetry requires the reverse of every ordered pair. Step 2: In option C, ((1,3)) is present but ((3,1)) is absent. Step 3: When a non-identical pair appears alone, check its reverse carefully.

Step 1: The reverse of ((2,2)) is the same pair. Step 2: ((3,5)) is matched by ((5,3)). Step 3: If all reverse pairs are present, the relation is symmetric.

Step 1: ((1,2)) and ((2,1)) are reverse pairs, so they satisfy symmetry. Step 2: The reverse of ((2,3)), which is ((3,2)), is missing. Step 3: For symmetry, ((a,a)) pairs are not required unless reflexivity is asked.

Step 1: Symmetry fails only when some ((a,b)) is present without ((b,a)). Step 2: In the empty relation, no pair is present, so the rule is not violated. Step 3: Remember that an empty relation is symmetric.

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

Step 1: The reverse of a pair like ((a,a)) is again ((a,a)). Step 2: All given pairs are of this form. Step 3: Relations containing only diagonal pairs are easy cases of symmetry.

Step 1: In a symmetric relation, the reverse of the given pair is required. Step 2: The reverse of ((6,9)) is ((9,6)). Step 3: Other pairs are not automatically guaranteed by symmetry.

Step 1: ((1,2)) is matched by ((2,1)). Step 2: ((3,1)) is matched by ((1,3)), and ((2,2)) reverses to itself. Step 3: When every reverse pair is present, the relation is symmetric.

Step 1: Both ((a,b)) and ((b,a)) are present. Step 2: No given pair is missing its reverse. Step 3: For symmetry, ((a,a)) and ((b,b)) are not necessary.

Step 1: For the non-identical pair ((1,2)), the reverse ((2,1)) is required. Step 2: Only option C has both pairs together. Step 3: While checking options, pair every non-identical ordered pair with its reverse.

Step 1: This relation contains only pairs where both entries are equal. Step 2: Reversing ((x,x)) gives ((x,x)) again. Step 3: Pairs with equal entries do not violate symmetry.

Step 1: For symmetry, if ((a,b)) is present, we check ((b,a)). Step 2: If (a+b) is even, then (b+a) is also even because changing order does not change the sum. Step 3: In rule-based relations, check whether the reversed pair still satisfies the rule.

Step 1: (a-b=0) means (a=b). Step 2: The pairs are like ((1,1),(2,2),(3,3)), whose reverse is the same pair. Step 3: Pairs with equal entries are safe for symmetry.

Step 1: The base idea of a symmetric relation is the reverse pair. Step 2: The condition is satisfied when ((a,b)) is accompanied by ((b,a)). Step 3: Conditions like ((a,a)) or ((b,b)) relate more to reflexivity, not symmetry.

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

Step 1: Reversing ((a,a)) gives ((a,a)) again. Step 2: Therefore such pairs satisfy the symmetry condition. Step 3: Pairs with equal coordinates do not create a problem for symmetry.

Step 1: To form the reverse pair, interchange the first and second entries. Step 2: Reversing ((8,5)) gives ((5,8)). Step 3: This small step is the key in symmetry questions.

Step 1: ((2,1)) is present for ((1,2)). Step 2: ((4,3)) is present for ((3,4)). Step 3: If every pair has its reverse, the relation is symmetric.

Step 1: ((1,2)) and ((2,1)) are reverse pairs. Step 2: The reverse of ((1,3)), which is ((3,1)), is not given. Step 3: Identifying incomplete pairs is the fastest way to check symmetry.

Step 1: The rule of symmetry directly applies to the reverse pair. Step 2: The required reverse pair for ((p,q)) is ((q,p)). Step 3: Do not assume ((p,p)) or ((q,q)) unless given or required by another condition.