Symmetric relation

Step 1: A correct example must contain the reverse of every pair. Step 2: In ({(1,2),(2,1)}), both pairs are reverses of each other. Step 3: Small examples help you remember the definition clearly.

Step 1: In a symmetric relation, whenever ((a,b)) is present, ((b,a)) must also be present. Step 2: Here ((1,2)) and ((2,1)) are both present, so the condition holds. Step 3: In exams, always check the reverse of every non-diagonal pair.

Step 1: For symmetry, we check reverse ordered pairs. Step 2: The reverse of ((3,4)), namely ((4,3)), is present, and ((5,5)) reverses to itself. Step 3: Pairs with equal components never break symmetry.

Step 1: Check the reverse of each ordered pair. Step 2: ((2,1)) exists for ((1,2)), and ((3,2)) exists for ((2,3)). Step 3: If all reverse pairs are present, the relation is symmetric.

Step 1: A pair without its reverse needs its reverse pair. Step 2: ((2,3)) is present but ((3,2)) is missing. Step 3: To make a relation symmetric, add the missing reverse pairs.

Step 1: In a symmetric relation, reversing the order keeps the pair in the relation. Step 2: The reverse of ((7,9)) is ((9,7)). Step 3: In direct questions, swap the positions to identify the required pair.

Step 1: A symmetric relation is based on the reverse-pair condition. Step 2: The main idea is that ((a,b)) implies ((b,a)). Step 3: Keep this definition separate from reflexive and transitive relations.

Step 1: The reverse of ((a,a)) is again ((a,a)). Step 2: All given pairs are of this form, so their reverses are already present. Step 3: A relation with only diagonal pairs is symmetric.

Step 1: Every pair with different components needs its reverse. Step 2: The reverse of ((4,6)), which is ((6,4)), is not given. Step 3: If even one reverse pair is missing, the relation is not symmetric.

Step 1: Symmetry fails only when a pair exists without its reverse. Step 2: In the empty relation, there is no pair, so the condition is not violated. Step 3: Remember that the empty relation is symmetric.

Step 1: The universal relation contains all possible ordered pairs. Step 2: Therefore, for every ((a,b)), the reverse ((b,a)) is also present. Step 3: Treat (A\times A) as symmetric in such questions.

Step 1: The reverse of ((1,2)) should be ((2,1)). Step 2: In the first option, ((2,1)) is missing, so it is not symmetric. Step 3: A single off-diagonal pair often breaks symmetry.

Step 1: A pair with equal components does not change after reversal. Step 2: Reversing ((4,4)) gives ((4,4)) again. Step 3: Such pairs do not require adding a new reverse pair.

Step 1: (a-b=0) means (a=b). Step 2: If ((a,b)) is present, then (a=b), so ((b,a)) also satisfies the same condition. Step 3: Relations based on equality are usually symmetric.

Step 1: Equality works the same in both directions. Step 2: If (a=b), then (b=a) is also true. Step 3: In equality-based relations, symmetry is immediate.

A relation R on a set is symmetric when reversing the order of every related pair still gives a pair in the relation. Thus, if (a,b) belongs to R, then (b,a) must also belong to R. Option A states exactly this definition. Options B and C concern diagonal pairs and do not define symmetry, while D introduces an unrelated pair without requiring the reverse pair.

Step 1: ((1,3)) and ((3,1)) are reverses of each other. Step 2: ((2,2)) and ((3,3)) are their own reverses. Step 3: All pairs are covered, so the relation is symmetric.

Step 1: First find the pair that has no reverse. Step 2: ((1,3)) is present but its reverse ((3,1)) is missing. Step 3: Adding the missing reverse pair is the direct way to make a relation symmetric.

To test symmetry, inspect every non-diagonal ordered pair and check whether its reverse is also present. In option A, (1,2) is accompanied by (2,1), and (3,3) is its own reverse, so the relation is symmetric. Option B lacks (2,1) and (3,2); C lacks (3,1) and (1,2); D lacks (3,2). Therefore only A satisfies the definition.

Step 1: The rule of symmetry applies directly to the reverse pair. Step 2: From ((p,q)), ((q,p)) must be present. Step 3: This does not force ((p,p)) or ((q,q)) to be present.

Step 1: Swapping (x) and (y) in (x+y=5) gives (y+x=5). Step 2: Addition is unchanged by order, so the reverse pair also satisfies the condition. Step 3: Conditions based on equal addition are often symmetric.

Step 1: For symmetry, ((y,x)) must satisfy the same condition whenever ((x,y)) does. Step 2: If (x-y=2), then generally (y-x=2) is not true. Step 3: Be careful with subtraction-based conditions because order matters.

Step 1: In a symmetric relation, swapping order should not change the condition. Step 2: (a+b) and (b+a) are equal, so the condition of being even holds both ways. Step 3: Addition-based conditions are not affected by order.

Step 1: Look at the condition that (a+b) is even. Step 2: If (a+b) is even, then (b+a) is also even because changing order does not change the sum. Step 3: In sum-based conditions, reverse pairs are easy to verify.

Step 1: ((2,6)) and ((6,2)) are reverses of each other. Step 2: Such a pair supports the symmetry condition. Step 3: Remember, to call the whole relation symmetric, every pair must be checked.