Relations on Rational Numbers and Sets

Relation R on Q

• Relation R on the set of rational numbers (Q) is defined as: R = {(x, y) ∈ Q² | x ≤ y} \ {(0, 0), (0, 1)}
• This means x ≤ y, but (x, y) is not (0, 0) or (0, 1).

Counterexamples for R

• (a) R is not reflexive:

  • Reflexivity means (x, x) ∈ R for all x in Q.
  • For example, (0, 0) ∉ R by the definition, therefore R is not reflexive.

• (b) R is not symmetric:

  • Symmetry means if (x, y) ∈ R, then (y, x) ∈ R.
  • Consider (1, 2) ∈ R because 1 ≤ 2. (2, 1) ∉ R because 2 > 1. R is not symmetric.

• (c) R is not transitive:

  • Transitivity means if (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R.
  • Example: (1, 2) ∈ R and (2, 3) ∈ R, but 1 ≤ 3. (1, 3) ∈ R. Therefore R is not transitive.

Relations on A = {1, 2, 3, 4}

• (a) R is reflexive and transitive but not symmetric:

  • Example: R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}
  • R is reflexive because all elements relate to themselves.
  • R is transitive because if (a, b) and (b, c), then (a, c) is present.
  • R is not symmetric because if (a, b) exists, (b, a) might or might not exist. This example demonstrates that.

• (b) R is symmetric but neither reflexive nor transitive:

  • Example: R = {(1, 2), (2, 1), (2, 3), (3, 2)}
  • R is symmetric because if (a, b) ∈ R, then (b, a) ∈ R.
  • R is not reflexive because some elements are not related to themselves.
  • R is not transitive because (1, 2) ∈ R and (2, 3) ∈ R but (1, 3) is not present.