Symmetric relation

Step 1: (R) may contain both reverse pairs. Step 2: (S) can remove only one of those pairs, breaking symmetry. Step 3: Be careful with difference of relations because symmetry is not always preserved.

Step 1: The relation is already symmetric because every non-diagonal pair has its reverse. Step 2: To make it reflexive, ((1,1),(2,2),(3,3)) must be added. Step 3: Check symmetry and reflexivity separately; do not mix the two properties.

Step 1: ((1,2)) and ((2,1)) form a complete reverse pair. Step 2: ((2,4)) and ((4,2)) also form a complete pair. Step 3: Only ((3,4)) is missing its reverse ((4,3)), so one pair is needed.

Step 1: If (a^2=b^2), then (b^2=a^2) is also certainly true. Step 2: Hence ((a,b)) implies ((b,a)). Step 3: Conditions based on equality are often symmetric, while inequalities or divisibility need separate checking.

Step 1: Symmetry would require (a\mid b) to imply (b\mid a). Step 2: (2\mid4) is true, but (4\nmid2). Step 3: A small divisibility counterexample is very useful in exams.

Step 1: In a symmetric relation, every given pair needs its reverse. Step 2: The reverse of ((1,2)) is ((2,1)), and the reverse of ((2,3)) is ((3,2)). Step 3: Symmetry does not force extra diagonal pairs.

Step 1: Having every ((a,a)) is related to reflexivity. Step 2: Symmetry only says that the reverse of every present pair must also be present. Step 3: Keep reflexive and symmetric properties separate in exams.

Step 1: (|a-b|=|b-a|). Step 2: Therefore, if (|a-b|<2), then (|b-a|<2) too. Step 3: For absolute difference relations, check this equality first.

Step 1: If (a+b=10), then (b+a=10) is also true. Step 2: Thus the reverse pair ((b,a)) is also in the relation. Step 3: For sum-based conditions, use the commutative nature of addition.

Step 1: For ((a,b)) to be in the relation, (a=2b) is required. Step 2: The reverse pair ((b,a)) would require (b=2a), which is not generally true. Step 3: ((2,1)) is a simple counterexample because ((1,2)) is not in the relation.

Step 1: For (n=3), the total independent choices are (\frac{3\cdot4}{2}=6). Step 2: Including ((1,2)) forces ((2,1)), so one non-diagonal block is fixed. Step 3: The remaining independent choices are (4), so the number is (2^4=16).

Step 1: For (n=3), symmetric relations have (6) independent choices. Step 2: Excluding ((1,1)) fixes one of these choices. Step 3: The remaining (5) choices are free, so the number is (2^5=32).

Step 1: Symmetry talks only about pairs that are present. Step 2: If a pair is absent, we cannot always conclude anything definite about its reverse. Step 3: In exams, avoid reversing the implication incorrectly.

Step 1: A symmetric relation (R) may contain pairs together with their reverses. Step 2: A subset (S) may keep only one pair and remove the reverse. Step 3: Therefore, every subrelation of a symmetric relation need not be symmetric.

Step 1: (a\equiv b \pmod{2}) means both numbers have the same parity. Step 2: If (a) and (b) have the same parity, then (b) and (a) also have the same parity. Step 3: In such congruence relations, reversing the order does not change the condition.

Step 1: For a symmetric relation, (R^{-1}=R). Step 2: Hence (R\cup R^{-1}=R\cup R=R). Step 3: In inverse-relation questions, first identify whether (R^{-1}=R).

The governing concept is symmetry: a relation S is symmetric when (a,b) ∈ S implies (b,a) ∈ S. If (a,b) belongs to R, then (b,a) belongs to R⁻¹; if (a,b) belongs to R⁻¹, then (b,a) belongs to R. Thus every pair in R ∪ R⁻¹ brings its reverse pair into the same union. Therefore option A is always true. Reflexivity and transitivity are not guaranteed, and the union need not be empty.

Step 1: (R^{-1}) contains ((2,1)) from ((1,2)) and ((3,2)) from ((2,3)). Step 2: The union keeps both original and reverse pairs. Step 3: This is the symmetric closure of (R).

Step 1: In a relation matrix, (m_{ij}=1) means ((a_i,a_j)\in R). Step 2: Symmetry requires ((a_j,a_i)), so (m_{ji}=1). Step 3: Therefore, the matrix of a symmetric relation is a symmetric matrix.

Step 1: (m_{24}=1) means ((a_2,a_4)\in R). Step 2: Symmetry would require (m_{42}=1), but it is (0). Step 3: One unequal reflected entry breaks symmetry.