Symmetric relation

Step 1: In a symmetric relation, ((4,1)\in R) would force ((1,4)\in R). Step 2: Since ((1,4)\notin R), ((4,1)) cannot be in (R). Step 3: In symmetric relations, absence of an off-diagonal pair also implies absence of its reverse.

Step 1: (a\equiv b \pmod{3}) means (a) and (b) have the same remainder on division by (3). Step 2: If (a) and (b) have the same remainder, then (b) and (a) also have the same remainder. Step 3: For same-remainder relations, check the reversed order mentally.

Step 1: Test one pair for symmetry. ((2,1)) belongs because (2\equiv 2\cdot1 \pmod{5}). Step 2: The reverse ((1,2)) would require (1\equiv4 \pmod{5}), which is false. Step 3: Modular relations with a multiplier do not always survive reversal.

Step 1: There are (3) ways to choose one diagonal pair. Step 2: Two off-diagonal ordered pairs mean one reverse-pair group; with (3) elements, there are (\binom{3}{2}=3) such groups. Step 3: Total possibilities are (3\cdot3=9).

Step 1: The difference of two symmetric relations is also symmetric. Step 2: Both (R-S) and (S-R) are symmetric, and the union of symmetric relations is symmetric. Step 3: Break a complex operation into smaller relation operations.

Step 1: (\Delta) contains only diagonal pairs of the form ((a,a)). Step 2: The reverse of such a pair is the same pair, so (\Delta) is symmetric and (R\cap\Delta) remains symmetric. Step 3: The diagonal relation is always easy to handle in symmetry questions.

Step 1: Symmetry only says that every existing pair must have its reverse. Step 2: It does not require diagonal pairs like ((a,a)) to be present. Step 3: Treat symmetry and reflexivity as separate properties.

Step 1: From ((a,b)\in R), symmetry gives ((b,a)\in R). Step 2: By transitivity, ((a,b)) and ((b,a)) imply ((a,a)\in R). Step 3: This conclusion applies to the involved elements, not automatically to every element of the set.

Step 1: Symmetry says that if ((a,b)) is present, then ((b,a)) is present. Step 2: Antisymmetry says that if both directions are present, then (a=b). Hence off-diagonal pairs with (a\ne b) cannot occur. Step 3: When both properties appear together, separate diagonal and off-diagonal cases.

Step 1: If a relation is both symmetric and antisymmetric, no off-diagonal pair can be present. Step 2: Only the (4) diagonal pairs can be chosen independently. Step 3: Therefore, the number of such relations is (2^4).

Step 1: If (xy>0), then (yx>0) because the product does not change when order is reversed. Step 2: Hence ((x,y)\in R) implies ((y,x)\in R). Step 3: In product-based conditions, check whether swapping changes the value.

Step 1: ((2,1)) belongs to the relation because (2=2\cdot1). Step 2: The reverse ((1,2)) does not belong because (1\ne2\cdot2). Step 3: One valid counterexample proves that the relation is not symmetric.

Step 1: Since (R) is symmetric, (R^{-1}=R). Step 2: ((R\circ R)^{-1}=R^{-1}\circ R^{-1}=R\circ R), so (R^2) equals its inverse. Step 3: For powers of relations, the inverse relation rule is very useful.

Step 1: Since (R) is symmetric, (R^{-1}=R). Step 2: ((R^3)^{-1}=(R\circ R\circ R)^{-1}=R^{-1}\circ R^{-1}\circ R^{-1}=R^3). Step 3: Powers of a symmetric relation remain symmetric.

Step 1: The symmetry condition applies only to pairs that are present. Step 2: The empty relation has no pair, so there is no violation of symmetry. Step 3: Such empty conditions are treated as true in mathematics.

Step 1: (A\times A) contains all possible ordered pairs from (A). Step 2: If ((a,b)) is present, then ((b,a)) is also definitely present. Step 3: The universal relation is one of the simplest examples of a symmetric relation.

Step 1: Each off-diagonal pair group contributes (2) ordered pairs. Step 2: (3) groups give (6) off-diagonal pairs, and with (2) diagonal pairs the total is (8). Step 3: Remember the difference between pair groups and actual ordered pairs.

Step 1: (\min(a,b)=a) means (a\le b). Step 2: ((1,2)) is in the relation, but ((2,1)) is not. Step 3: In such questions, first convert the condition into a familiar inequality.

Step 1: (\max(a,b)=\max(b,a)) is true for all real (a,b). Step 2: So the relation contains every ordered pair, making it universal. Step 3: A universal relation is always symmetric because every reverse pair is present.

Step 1: For distance, (|a-b|=|b-a|). Step 2: If (|a-b|<5), then (|b-a|<5) also holds. Step 3: Distance-based relations do not have direction, so they are often symmetric.

Step 1: ((1,2)) and ((2,1)) form one pair group. Step 2: ((3,4)) and ((4,3)) form the second pair group. Step 3: In symmetric counting, reverse pairs are not counted as separate groups.

Step 1: The larger relation (S) may contain additional pairs. Step 2: If an extra pair is added without its reverse, (S) may fail to be symmetric. Step 3: A property need not pass upward to every superset relation.

Step 1: Both ((1,2)) and ((2,1)) are present, and diagonal pairs are their own reverses. Step 2: Hence the relation is symmetric, but it is not reflexive because ((3,3)) is missing. Step 3: Symmetry does not require every diagonal pair to be present.

Step 1: (R^{-1}\subseteq R) means all pairs of the inverse relation are contained in (R). Step 2: If ((a,b)\in R), then ((b,a)\in R^{-1}), so by inclusion ((b,a)\in R). Step 3: Inclusion of inverse relations is a quick way to prove symmetry.

Step 1: In a symmetric relation, if ((a,b) \in R), then ((b,a) \in R) must also be present. Step 2: Here ((1,2)) has ((2,1)), and ((2,3)) has ((3,2)). Step 3: In exams, always check reverse pairs for pairs with different elements.