Symmetric relation

Step 1: ((1,2)) and ((2,1)) already form a reverse pair. Step 2: ((1,3)) is present, but its reverse ((3,1)) is missing. Step 3: To make a relation symmetric, add only the missing reverse pairs.

Step 1: ((1,4)) with ((4,1)) is complete, and ((2,3)) with ((3,2)) is also complete. Step 2: The reverse of ((1,2)), namely ((2,1)), is missing. Step 3: Adding just ((2,1)) makes the relation symmetric.

Step 1: If (|a|=|b|), then the reversed equality (|b|=|a|) is also true. Step 2: Hence (aRb) implies (bRa). Step 3: Equality of absolute values gives symmetry directly.

Step 1: If (a\ne b), then (b\ne a) is also true. Step 2: Hence whenever ((a,b)) is in the relation, ((b,a)) is also in it. Step 3: The not-equal condition does not change when the order is reversed.

Step 1: If (a+b) is odd, then (b+a) is also odd. Step 2: Changing the order of addition does not change the sum or its parity. Step 3: Relations based on parity of a sum are usually easy symmetry checks.

Step 1: ((1,2)) is in the relation because (1<2). Step 2: The reverse ((2,1)) is not in the relation because (2<1) is false. Step 3: Order relations like less than or greater than are generally not symmetric.

Step 1: Diagonal pairs do not need a separate reverse pair. Step 2: ((1,2)) and ((2,1)) are complete, but ((1,3)) is missing its reverse ((3,1)). Step 3: Identifying the missing reverse pair is the key in such questions.

Step 1: There are (n) independent diagonal pairs. Step 2: Off-diagonal pairs ((a,b)) and ((b,a)) are chosen together, and there are (\frac{n(n-1)}{2}) such unordered pairs. Step 3: Total independent choices are (n+\frac{n(n-1)}{2}=\frac{n(n+1)}{2}).

Step 1: (R^{-1}) contains the reverse of all pairs of (R). Step 2: If (R=R^{-1}), then every pair has its reverse in the same relation, which is symmetry. Step 3: Treat (R=R^{-1}) as an alternative test for symmetry.

Step 1: If (a^2+b^2=17), then (b^2+a^2=17) also holds. Step 2: Changing the order of addition does not change the sum, so the reverse pair also belongs to the relation. Step 3: Conditions involving sums of squares are often symmetric because order does not matter.

Step 1: ((2,1)) is in the relation because (2=2\cdot1). Step 2: The reverse ((1,2)) is not in the relation because (1\ne2\cdot2). Step 3: In multiplier relations, reversing changes the ratio, so look for a counterexample.

Step 1: On a (3)-element set, there are (6) independent choices. Step 2: Including ((1,2)) forces ((2,1)), and excluding ((2,3)) also excludes ((3,2)); two independent choices are fixed. Step 3: Thus (4) independent choices remain, giving (2^4).

Step 1: The (4) diagonal pairs are compulsory, so their choices are fixed. Step 2: The number of independent off-diagonal unordered pairs is (\frac{4\cdot3}{2}=6). Step 3: Each of these (6) pairs can be chosen or omitted, so the answer is (2^6).

Step 1: All (4) diagonal pairs are fixed as absent. Step 2: Only the off-diagonal unordered pairs remain free, and their number is (\frac{4\cdot3}{2}=6). Step 3: Each of these (6) blocks can be included or excluded.

Step 1: If ((a-b)^2=4), then ((b-a)^2=(-(a-b))^2=4). Step 2: Hence the reverse pair also belongs to the relation. Step 3: Squaring removes the effect of the negative sign, giving symmetry.

Step 1: If (a+b\le6), then (b+a\le6) also holds. Step 2: Changing the order of addition does not change the sum, so the reverse pair is also included. Step 3: Even inequality conditions based on a sum can be symmetric.

Step 1: ((4,1)) is in the relation because (4+2\cdot1=6). Step 2: The reverse ((1,4)) is not in the relation because (1+2\cdot4=9). Step 3: When coefficients of (a) and (b) differ, reversal often changes the condition.

Step 1: A symmetric larger relation may contain both a pair and its reverse. Step 2: A subrelation can choose only one of them, breaking symmetry. Step 3: Symmetry of a subrelation must be checked separately.

Step 1: (R) contains both ((1,2)) and ((2,1)). Step 2: (S) contains both ((2,3)) and ((3,2)), so the union keeps all reverse pairs. Step 3: The union of symmetric relations remains symmetric.

Step 1: Every off-diagonal pair has its reverse already present in the relation. Step 2: ((3,3)) reverses to itself, so the inverse relation contains the same pairs. Step 3: If (R^{-1}=R), the relation is symmetric.