Symmetric relation

Step 1: For (4) elements, there are (10) independent groups. Step 2: The groups containing ((1,2)) and ((3,4)) are fixed as included. Step 3: The remaining (8) groups are free, so the number is (2^8).

Step 1: A symmetric relation must contain ((2,1)) whenever it contains ((1,2)). Step 2: The condition asks for ((1,2)) to be present but ((2,1)) absent, which violates symmetry. Step 3: Hence the number of such relations is (0).

Step 1: ((a_i,a_j)) in a relation means (m_{ij}=1). Step 2: Symmetry requires ((a_j,a_i)), meaning (m_{ji}=1). Step 3: Thus (m_{ij}=m_{ji}) for all (i,j), so (M=M^T).

Step 1: ((1,2)) needs ((2,1)). Step 2: ((2,3)) needs ((3,2)), and ((3,1)) needs ((1,3)). Step 3: All three reverse pairs are missing, so the minimum number to add is (3).

Step 1: ((1,2)) and ((2,1)) already appear as a reverse pair. Step 2: The reverse of ((3,4)), namely ((4,3)), is missing. Step 3: Adding only ((4,3)) is enough and minimal.

Step 1: (a-b=0) means (a=b). Step 2: If (a=b), then (b=a), so the reverse pair also belongs. Step 3: Equality-based relations are symmetric, while conditions like (a-b=1) change when reversed.

Step 1: Symmetry is about the presence of reverse pairs. Step 2: If ((a,b)\in R), then ((b,a)\in R) must also hold. Step 3: The diagonal condition is reflexivity, and the three-pair condition is transitivity.

Step 1: The difference of two symmetric relations is symmetric. Step 2: Therefore (R-S) and (S-R) are both symmetric. Step 3: The union of two symmetric relations is symmetric, so (R\triangle S) is symmetric.

Step 1: (a\equiv -b \pmod{5}) means (a+b\equiv0 \pmod{5}). Step 2: Reversing the order gives (b+a\equiv0 \pmod{5}), the same condition. Step 3: Therefore the reverse pair also belongs, so the relation is symmetric.

Step 1: ((1,2)) and ((2,1)) are already balanced, and diagonal pairs cause no issue. Step 2: ((2,3)) is present but ((3,2)) is missing. Step 3: If only removal is allowed, remove ((2,3)), so the minimum number is (1).

Step 1: A relation is symmetric if ((a,b)\in R) implies ((b,a)\in R). Step 2: Here the condition is (a+b\text{ is even}), and (a+b=b+a), so the reverse pair also satisfies it. Step 3: In exams, check commutative conditions like addition first.

Step 1: If (a-b) is divisible by (2), then (b-a=-(a-b)) is also divisible by (2). Step 2: So ((a,b)\in R) implies ((b,a)\in R). Step 3: Divisibility conditions often remain symmetric after changing the sign.

Step 1: The reverse of ((1,2)) is ((2,1)), and both are present. Step 2: The reverse of ((2,3)) is ((3,2)), and ((1,1)) is its own reverse. Step 3: Symmetry does not require all possible pairs, only reverse pairs of the chosen pairs.

Step 1: In a symmetric relation, every included ordered pair needs its reverse. Step 2: ((a,b)) requires ((b,a)), and ((c,a)) requires ((a,c)). Step 3: Diagonal pairs are required for reflexivity, not for symmetry alone.

Step 1: The number of symmetric relations on a set with (n) elements is (2^{\frac{n(n+1)}{2}}). Step 2: For (n=3), the exponent is (\frac{3\cdot4}{2}=6). Step 3: Diagonal pairs are chosen independently, while non-diagonal pairs are chosen in reverse-pair blocks.

Step 1: For four elements, the number of non-diagonal reverse-pair blocks is (\frac{4\cdot3}{2}=6). Step 2: All diagonal pairs are compulsory, so only these six blocks can be included or excluded. Step 3: When reflexive and symmetric are combined, do not count diagonal choices as free.

Step 1: There are (n) diagonal positions and (\frac{n(n-1)}{2}) non-diagonal reverse-pair blocks. Step 2: Total independent choices become (n+\frac{n(n-1)}{2}=\frac{n(n+1)}{2}). Step 3: While counting symmetric relations, do not treat ((a,b)) and ((b,a)) as independent choices.

Step 1: A relation is symmetric when its matrix is symmetric about the main diagonal. Step 2: Here (m_{13}=m_{31}=1), (m_{24}=m_{42}=1), and the other matching positions also agree. Step 3: In matrix questions, compare entries across the main diagonal.

Step 1: A symmetric relation has a matrix satisfying (m_{ij}=m_{ji}). Step 2: In the first matrix, all corresponding entries across the diagonal are equal. Step 3: Even one mismatched pair is enough to make the relation non-symmetric.

Step 1: In an inverse relation, every ordered pair is reversed. Step 2: In a symmetric relation, the reversed pair is already in the same relation, so (R^{-1}=R). Step 3: Remembering the link between symmetry and inverse relation saves time in exams.