Relations on Rational Numbers and Sets

Study Notes

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.

Description

Explore the properties of relation R on the set of rational numbers and the defined set A. This quiz covers reflexivity, symmetry, and transitivity, providing examples and counterexamples for each property. Test your understanding of these fundamental concepts in relation theory!