Symmetric relation

Step 1: For symmetry, we only need to check reverse pairs of the pairs that are present. Step 2: ((1,2)) has ((2,1)), and ((2,3)) has ((3,2)). A diagonal pair reverses to itself. Step 3: All diagonal pairs are not required for symmetry.

Step 1: ((1,2)) is in the relation because (1\le2). Step 2: Its reverse ((2,1)) is not in the relation because (2\le1) is false. Step 3: To disprove symmetry, show a present pair whose reverse is absent.

Step 1: The definition of a symmetric relation says that ((b,a)) must be present whenever ((a,b)) is present. Step 2: Since ((4,7)) is given, ((7,4)) must be present. Step 3: Symmetry does not automatically force diagonal pairs.

A symmetric relation is determined by independent choices on the diagonal and on unordered pairs of distinct elements. For n elements, the number of such choices is n + n(n−1)/2 = n(n+1)/2. With n = 5, the exponent is 5×6/2 = 15. Each choice is binary: include the corresponding diagonal pair, or include both members of an off-diagonal pair; alternatively, omit it. Therefore the number of symmetric relations is 2¹⁵, so option A is correct. The exponents 20 and 25 do not count the required paired positions.

Step 1: There are (6) independent choices for symmetric relations on a (3)-element set. Step 2: If ((1,2)) is included, symmetry forces ((2,1)), so one off-diagonal block is fixed. Step 3: Therefore (4) independent choices remain, giving (2^4).

On a three-element set, a symmetric relation has 3 diagonal choices and 3 unordered off-diagonal choices, for a total of 6 independent binary choices. The condition that (1,1) is absent fixes one diagonal choice: it must be excluded rather than chosen. The remaining diagonal pairs (2,2) and (3,3), together with the three off-diagonal pair groups, remain independently selectable. Thus five binary choices remain, giving 2⁵ symmetric relations. Option A is correct; 2⁶ would allow (1,1), while the smaller powers omit valid choices.

Step 1: (a \equiv b \pmod{2}) means (a-b) is divisible by (2). Step 2: Then (b-a=-(a-b)) is also divisible by (2), so the reverse pair also belongs to the relation. Step 3: Same-remainder relations are usually symmetric.

Step 1: If (ab=0), then (ba=0) also holds. Step 2: Multiplication is commutative, so the reversed pair also satisfies the condition. Step 3: Commutativity of addition or multiplication is very useful in checking symmetry.

Step 1: ((2,1)) is in the relation because (2-1=1). Step 2: Its reverse ((1,2)) is not in the relation because (1-2=-1). Step 3: For fixed-difference relations, reversing changes the sign, so be careful.

Step 1: If ((a,b) \in R \cap S), then ((a,b)) is in both (R) and (S). Step 2: Since both are symmetric, ((b,a)) is in both, so ((b,a) \in R \cap S). Step 3: Intersection preserves the reverse-pair condition.

Step 1: If ((a,b) \in R \cup S), then it belongs to (R) or (S). Step 2: The relation containing it is symmetric, so ((b,a)) also belongs to that relation and hence to the union. Step 3: The union of symmetric relations is also symmetric.

Step 1: (R^{-1}) contains the reverse of every ordered pair in (R). Step 2: Symmetry means every reverse pair is already in (R), so the inverse relation equals (R). Step 3: Remember symmetry as (R=R^{-1}).

Symmetry is defined by reversal of ordered pairs. A relation R is symmetric when, for every a and b in its underlying set, the presence of (a,b) in R guarantees the presence of (b,a) in R. Option A states this condition directly and is therefore correct. Option B requires every self-pair and defines reflexivity, not symmetry. Option C requires completion of a two-step chain and defines transitivity. Option D describes the empty relation, which is indeed symmetric but is not the definition of all symmetric relations; many non-empty relations are symmetric as well. Thus only option A gives the general definition rather than a special example or a different property.

Step 1: The symmetry condition applies only to pairs that are present in the relation. Step 2: The empty relation has no pair, so no pair violates the condition. Step 3: Remember that the empty relation is symmetric by vacuous truth.

Step 1: (A\times A) contains all possible ordered pairs from (A). Step 2: If ((a,b)) is in it, then (a,b \in A), so ((b,a)) is also in (A\times A). Step 3: The universal relation is always symmetric.

Step 1: Check the reverse of each off-diagonal pair. Step 2: ((1,2)) has ((2,1)), and ((3,4)) has ((4,3)). ((2,2)) reverses to itself. Step 3: A symmetric relation need not contain every ((a,a)) pair.

Step 1: If (a+b=0), then (b+a=0) also holds. Step 2: Addition is commutative, so the reversed pair satisfies the condition. Step 3: Symmetry is about the reverse pair, not necessarily equality of the two elements.

Step 1: (\gcd(a,b)=\gcd(b,a)). Step 2: So if the greatest common divisor of ((a,b)) is (2), the same is true for ((b,a)). Step 3: Quantities unchanged by swapping often define symmetric relations.

Step 1: (\operatorname{lcm}(a,b)=\operatorname{lcm}(b,a)). Step 2: Therefore, if a pair has least common multiple (6), its reverse also has least common multiple (6). Step 3: Both LCM and GCD remain unchanged when the order is swapped.

Step 1: Symmetry only demands reverse pairs for pairs that are present. Step 2: Having every ((a,a)) is the condition for reflexivity, not symmetry. Step 3: Keeping definitions separate helps solve such trap questions.