Symmetric relation

Correct answer: A. Every symmetric relation is reflexive

The key distinction is between symmetry and reflexivity. Symmetry requires that (a,b) in R imply (b,a) in R. Reflexivity requires that (a,a) belong to R for every a in the underlying set. These conditions are independent: the empty relation is symmetric because it has no counterexample, but on a non-empty set it is not reflexive. The universal relation is both symmetric and reflexive, and the identity relation is also symmetric because each pair (a,a) equals its own reverse. Since a symmetric relation can lack all self-pairs, the claim that every symmetric relation is reflexive is false. Therefore option A is correct.

Correct answer: A. Because there is no pair that violates the condition

Step 1: Symmetry fails only when some ((a,b)) is present but ((b,a)) is absent. Step 2: The empty relation has no pair, so there is no violation. Step 3: This is an example of a vacuously true statement.

Correct answer: A. Because every possible pair and its reverse are present

Step 1: The universal relation contains all ordered pairs from (A\times A). Step 2: Therefore, whenever ((a,b)) is present, ((b,a)) is also present. Step 3: Having all pairs makes symmetry automatic.

Correct answer: A. It is symmetric

Step 1: The identity relation contains only pairs of the form ((a,a)). Step 2: The reverse of such a pair is the same pair ((a,a)). Step 3: Hence the identity relation is always symmetric.

Correct answer: A. It is symmetric

Step 1: (a\equiv b \pmod{2}) means both have the same parity. Step 2: If (a) has the same parity as (b), then (b) has the same parity as (a). Step 3: Congruence-type conditions are usually symmetric.

Correct answer: A. Because congruence remains true after swapping the order

Step 1: (a\equiv b \pmod{3}) means (a) and (b) have the same remainder when divided by 3. Step 2: Having the same remainder works in both directions. Step 3: Swapping the order does not break congruence.

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

Step 1: Both ((1,2)) and ((2,1)) are present, so the relation is symmetric. Step 2: But ((1,1)) and ((2,2)) are missing, so it is not reflexive. Step 3: Symmetry and reflexivity are separate properties.

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

Step 1: ((1,1)) and ((2,2)) are present, so the relation is reflexive. Step 2: ((1,2)) is present but ((2,1)) is missing, so it is not symmetric. Step 3: In mixed-property questions, test each property separately.

Correct answer: A. ((3,1))

Step 1: ((1,2)) and ((2,1)) are present. Step 2: ((2,3)) and ((3,2)) are also present, but the reverse of ((1,3)), namely ((3,1)), is missing. Step 3: Add only the missing reverse pair.

Correct answer: A. ((2,1)) and ((3,2))

Step 1: Symmetry requires the reverse of each given pair. Step 2: The reverse of ((1,2)) is ((2,1)), and the reverse of ((2,3)) is ((3,2)). Step 3: For a minimum completion, add only the required reverse pairs.

Correct answer: A. Yes, because (ab=ba)

Step 1: If (ab) is even, then (ba) is also even because multiplication is commutative. Step 2: Hence the reverse pair also belongs to the relation. Step 3: For product-based conditions, check what happens after swapping.

Correct answer: A. Symmetric

Step 1: If (a+b=5), then (b+a=5) as well. Step 2: Therefore the reverse of every related pair also belongs to the relation. Step 3: The key signal is that changing the order does not change the sum.

Correct answer: A. Because ((3,1)\in R) but ((1,3)\notin R)

Step 1: ((3,1)) belongs to the relation because (3-1=2). Step 2: Its reverse ((1,3)) does not satisfy (1-3=2). Step 3: A fixed directed difference usually breaks symmetry.

Correct answer: A. (R={(a,b):a>b})

Step 1: If (a>b), the reversed condition (b>a) is generally false. Step 2: The other conditions remain true after swapping. Step 3: Directional comparisons are a common source of non-symmetry.

Correct answer: A. ((5,2)) and ((7,5)) are in (R)

Step 1: Symmetry guarantees only the reverse pairs of the given pairs. Step 2: ((2,5)) gives ((5,2)), and ((5,7)) gives ((7,5)). Step 3: Symmetry does not create a new connecting pair like ((2,7)).

Correct answer: A. ((a,a)\in R) for every (a\in A)

Step 1: Symmetry is based on reverse ordered pairs. Step 2: Having every diagonal pair is the condition for reflexivity, not symmetry. Step 3: Mixing property names is a common exam mistake.

Correct answer: A. (R) cannot be symmetric

Step 1: For ((1,2)), the reverse pair ((2,1)) is required. Step 2: Since it is missing, the definition of symmetry fails. Step 3: One counterexample is enough to disprove symmetry.

Correct answer: A. (M=M^T)

Step 1: In a symmetric relation, the presence of ((i,j)) and ((j,i)) must match. Step 2: In matrix form, this means (m_{ij}=m_{ji}). Step 3: This condition is written as (M=M^T).

Correct answer: A. (A={1,2,3}, R={(1,2),(2,1),(2,3),(3,2)})

Step 1: In the given (R), every pair has its reverse, so it is symmetric. Step 2: ((1,2)) and ((2,3)) are present, but ((1,3)) is missing, so transitivity fails. Step 3: Symmetry does not automatically imply transitivity.

Correct answer: A. Absence of both ((2,3)) and ((3,2))

Step 1: Symmetry fails when a present pair lacks its reverse. Step 2: If both ((2,3)) and ((3,2)) are absent, there is no imbalance. Step 3: Absence in both directions does not break symmetry.

Correct answer: A. Because each pair is its own reverse

Step 1: Reversing a diagonal pair ((a,a)) gives the same pair ((a,a)). Step 2: So the reverse of each such pair is already present. Step 3: Diagonal pairs are always safe for symmetry.

Correct answer: A. Symmetric

Step 1: If (a) and (b) are both even or both odd, the same remains true after swapping them. Step 2: Therefore ((a,b)) implies ((b,a)). Step 3: Same-type conditions often give symmetric relations.

Correct answer: A. It is symmetric

Step 1: If (a^2=b^2), then reversing the equality gives (b^2=a^2). Step 2: So ((a,b)) implies ((b,a)). Step 3: Equality-based conditions are easy to test by swapping sides.

Correct answer: A. If (a) is a friend of (b), then (b) is also a friend of (a)

Step 1: Symmetry means the relation works in both directions. Step 2: For friendship to be symmetric, if one person is a friend of another, the second must also be a friend of the first. Step 3: Even in real-life examples, check the reverse direction.

Correct answer: A. It is both symmetric and reflexive

Step 1: Both ((1,2)) and ((2,1)) are present, so symmetry holds. Step 2: ((1,1),(2,2),(3,3)) are all present, so reflexivity also holds. Step 3: When two properties are asked, verify both separately.