The five properties of relations are:
1. Reflexive: Every element is related to itself.
2. Symmetric: If x is related to y, then y is related to x.
3. Transitive: If x is related to y and y is related to z, then x is related to z.
4. Anti-symmetric: If x is related to y and y is related to x, then x = y.
5. Asymmetric: If x is related to y, then y is not related to x.
Using these properties, we can analyze the given relations as follows:
R1 = (x,y)
- Reflexive: Yes, because x divides x for any x.
- Symmetric: No, because if x divides y, it is not necessarily true that y divides x.
- Transitive: Yes, because if x divides y and y divides z, then x divides z.
- Anti-symmetric: No, because x can be a proper divisor of y.
- Asymmetric: No, because if x divides y, then y is not necessarily not related to x.
R2 = (x,y)
- Reflexive: Yes, because x + x = 2x is even for any x.
- Symmetric: Yes, because if x + y is even, then y + x is also even.
- Transitive: Yes, because if x + y and y + z are even, then x + z is even.
- Anti-symmetric: No, because x + y = y + x can hold for distinct values of x and y.
- Asymmetric: No, because if x + y is even, then y + x is not necessarily not related to x.
R3 = xy is even
- Reflexive: No, because 0 is not in N and any other element multiplied by 0 is 0, which is not even.
- Symmetric: Yes, because if xy is even, then yx is also even.
- Transitive: Yes, because if xy and yz are even, then xz is even.
- Anti-symmetric: No, because if xy = 0 and x and y are both non-zero, then x and y are distinct and both related to 0.
- Asymmetric: No, because if xy is even, then yx is not necessarily not related to x.
S1 = (2,y)
- Reflexive: No, because 2 divides 0, which is not in N.
- Symmetric: No, because if y divides z, then z does not necessarily divide y.
- Transitive: Yes, because if y divides z and z divides w, then y divides w.
- Anti-symmetric: No, because 2 can divide distinct y values.
- Asymmetric: No, because if y divides z, then z does not necessarily not related to y.
S2 = (2,y)
- Reflexive: No, because 2 + 2 = 4 is even, not odd.
- Symmetric: Yes, because if x + y is odd, then y + x is also odd.
- Transitive: Yes, because if x + y and y + z are odd, then x + z is even and not related.
- Anti-symmetric: Yes, because if x + y and y + x are odd, then x = y.
- Asymmetric: Yes, because if x + y is odd, then y + x is not related.
S3 = (2,y)
- Reflexive: No, because 2 multiplied by any odd number is even, not odd.
- Symmetric: Yes, because if xy is odd, then yx is also odd.
- Transitive: Yes, because if xy and yz are odd, then xz is odd.
- Anti-symmetric: No, because 2 can multiply distinct odd y values.
- Asymmetric: No, because if xy is odd, then yx is not necessarily not related to x.