Symmetric relation

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).

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.

Step 1: Here (n=5). Step 2: The number of symmetric relations is (2^{\frac{n(n+1)}{2}}). Step 3: (2^{\frac{5\cdot6}{2}}=2^{15}=32768), so the answer is (32768).

Step 1: The four diagonal pairs are fixed as included. Step 2: The number of non-diagonal reverse-pair blocks is (\frac{4\cdot3}{2}=6). Step 3: Each block has two choices, so the total number is (2^6=64).

Step 1: Having no diagonal pair means all four diagonal choices are fixed as excluded. Step 2: There are (\frac{4\cdot3}{2}=6) non-diagonal reverse-pair blocks. Step 3: These (6) blocks are freely chosen, so the number is (2^6=64).

Step 1: Symmetry is preserved under union and intersection, but not always under composition. Step 2: The ordered composition of two symmetric relations need not preserve all reverse pairs. Step 3: For composition questions, testing with a small counterexample is often best.

Step 1: Reversing (a) and (b) changes (a^2+b^2=1) to (b^2+a^2=1). Step 2: By commutativity of addition, this is the same condition. Step 3: For algebraic conditions, check whether the equation remains unchanged after swapping variables.

Step 1: (a^2-b^2=0) means (a^2=b^2). Step 2: Reversing gives (b^2=a^2), which is also true. Step 3: Simplifying an equation often makes checking symmetry easier.

Step 1: Symmetry requires the reverse of every present pair to be present. Step 2: ((2,5)) is present, but ((5,2)) is not. Step 3: This single missing reverse pair is enough to show the relation is not symmetric.

Step 1: ({(1,2),(2,1)}) contains both reverse pairs, so it is symmetric. Step 2: Reflexivity would require ((1,1),(2,2),(3,3)), which are missing. Step 3: Understanding the difference between symmetric and reflexive is very important.

Step 1: The first option contains all diagonal pairs, so it is reflexive. Step 2: It has ((1,2)) but not ((2,1)), so it is not symmetric. Step 3: In combined-property questions, check reflexivity and symmetry separately.

Step 1: A symmetric relation is based on the reverse-pair condition. Step 2: The option states exactly that if ((a,b)) is present, then ((b,a)) must be present. Step 3: Do not mix definitions; reflexive, symmetric, and transitive are separate properties.

Step 1: In a symmetric relation, if ((a,b)) is present, then ((b,a)) must also be present. Step 2: ((2,3)) is present but ((3,2)) is missing, while ((1,2)) and ((2,1)) already form a pair. Step 3: In exams, always check the reverse of every non-diagonal ordered pair.

Step 1: If (a+b) is even, then (b+a) is also even because addition is commutative. Step 2: Hence whenever ((a,b) \in R), ((b,a) \in R) also holds. Step 3: For such questions, reverse the condition and check whether it remains true.

Step 1: (a-b>0) means (a>b). Step 2: If (a>b), then generally (b>a) is false, so the reversed pair does not satisfy the condition. Step 3: In inequality-based relations, the reverse direction often fails.