Symmetric relation

Correct answer: A. Symmetric

Step 1: Find the reverses of pairs with different components. Step 2: ((2,1)) exists for ((1,2)), and ((4,2)) exists for ((2,4)). Step 3: ((3,3)) is automatically safe, so the relation is symmetric.

Correct answer: A. ((4,3))

Step 1: First form the reverse of each given pair. Step 2: ((1,2)) and ((2,1)) already form a pair, but ((4,3)) is missing for ((3,4)). Step 3: Identify the missing part through the reverse pair.

Correct answer: A. ((3,8)\in R)

Step 1: In a symmetric relation, the reversed pair must also be in the relation. Step 2: The reverse of ((8,3)) is ((3,8)). Step 3: Diagonal pairs are not required by this question.

Correct answer: A. ({(5,6),(6,5),(7,7)})

Step 1: Check the reverse of every non-diagonal pair. Step 2: In the first option, both ((5,6)) and ((6,5)) are present, while ((7,7)) reverses to itself. Step 3: To decide quickly, look for lone off-diagonal pairs.

Correct answer: A. Symmetric

Step 1: Same parity means both are even or both are odd. Step 2: If (a) and (b) have the same parity, then (b) and (a) also have the same parity. Step 3: Such two-way conditions show symmetry.

Correct answer: A. Because (b) need not be brother of (a)

Step 1: For symmetry, the relation must be true in both directions. Step 2: If (a) is brother of (b), (b) may be a sister, so (b) need not be brother of (a). Step 3: In real-life examples, read the meaning of the relation carefully.

Correct answer: A. Symmetric

Step 1: Being a classmate has the same meaning in both directions. Step 2: If (a) is a classmate of (b), then (b) is also a classmate of (a). Step 3: Connect two-way social relations with symmetry.

Correct answer: A. Symmetric

Step 1: (|a-b|) and (|b-a|) are equal. Step 2: So if (|a-b|=3), then (|b-a|=3) also holds. Step 3: Distance-like conditions using absolute value are often symmetric.

Correct answer: A. (a^2=b^2)

Step 1: For symmetry, the condition should remain true after swapping (a) and (b). Step 2: From (a^2=b^2), (b^2=a^2) is also true. Step 3: If equality remains true after swapping sides, symmetry holds.

Correct answer: A. (R) is symmetric

Step 1: The reverse of ((1,2)) is ((2,1)). Step 2: The reverse of ((2,3)) is ((3,2)), and ((1,1)) is fine by itself. Step 3: When all reverse checks pass, the relation is symmetric.

Correct answer: A. If the first is present, the second must also be present

Step 1: Symmetry requires the presence of the reverse order. Step 2: Therefore, if ((a,b)) is present, ((b,a)) must also be in the relation. Step 3: Remember this rule as the core definition.

Correct answer: A. ((6,6))

Step 1: Reversing an ordered pair swaps its two positions. Step 2: In ((6,6)), both components are equal, so the reverse is the same pair. Step 3: Pairs of the form ((a,a)) are always safe for symmetry.

Correct answer: A. 1

Step 1: Separate the pairs whose reverses already exist. Step 2: ((1,4)) and ((4,1)) are complete, and only ((5,2)) is needed for ((2,5)). Step 3: The number of missing reverse pairs gives the minimum number to add.

Correct answer: A. 2

Step 1: ((2,1)) is needed for ((1,2)). Step 2: ((4,3)) is needed for ((3,4)). Step 3: Two different reverse pairs are missing, so two pairs must be added.

Correct answer: A. ((a,a)\in R) must be present

Step 1: Symmetry guarantees only the reverse pair. Step 2: From ((a,b)), ((b,a)) must follow, but ((a,a)) need not. Step 3: Do not mix the reflexive condition with the symmetric condition.

Correct answer: A. ({(1,2),(2,1)}) on (A={1,2})

Step 1: First check symmetry: both ((1,2)) and ((2,1)) are present. Step 2: For reflexivity, ((1,1)) and ((2,2)) are also needed, which are absent here. Step 3: A symmetric relation need not be reflexive.

Correct answer: A. It is symmetric

Step 1: ((1,1)) and ((2,2)) are their own reverses. Step 2: Both ((1,2)) and ((2,1)) are present. Step 3: Therefore, every pair has its reverse in the relation.

Correct answer: A. ({(2,3),(3,2)})

Step 1: The reverse of ((2,3)) is ((3,2)). Step 2: The first option contains this required pair. Step 3: Other diagonal pairs do not fill the missing reverse pair.

Correct answer: A. Symmetric

Step 1: Living in the same city is a mutual condition. Step 2: If (a) lives in the same city as (b), then (b) lives in the same city as (a). Step 3: Recognize mutual same-property relations as symmetric.

Correct answer: A. (a) is greater than (b)

Step 1: Being greater than is a direction-based relation. Step 2: If (a>b), then (b>a) cannot be true. Step 3: Ordered comparison relations are usually not symmetric.

Correct answer: A. Symmetric

Step 1: The condition contains (a^2+b^2). Step 2: Swapping (a) and (b) gives (b^2+a^2=10), which is the same condition. Step 3: Addition keeps the condition symmetric after swapping.

Correct answer: A. Whether the reverse of every pair is in the relation

Step 1: Symmetry is checked by reverse pairs, not just by counting pairs. Step 2: For every ((a,b)), ((b,a)) should be found. Step 3: In exams, mark pairs in the list and match their reverses.

Correct answer: A. Yes, because both are their own reverses

Step 1: ((1,1)) and ((2,2)) do not change after reversal. Step 2: Symmetry requires reverses only for existing pairs, not all possible pairs. Step 3: Missing extra pairs do not automatically break symmetry.

Correct answer: A. 2

Step 1: ((1,2)) and ((2,1)) form one reverse pair. Step 2: ((3,1)) and ((1,3)) form another reverse pair. Step 3: Identifying reverse pairs helps in checking symmetry.

Correct answer: A. The reverse of some pair is not in the relation

Step 1: Lack of symmetry usually comes from a missing reverse pair. Step 2: If ((a,b)) is present but ((b,a)) is not, the relation is not symmetric. Step 3: Wrong options often reverse the correct definition.