Relations in Logic
Relations describe connections between two or more objects. Mathematically, a relation is a set of ordered pairs.
Example of a Relation
This relation says: R contains all pairs (x,y) where x is less than y.
Example: (2,5) ∈ R is true, because 2 < 5. But (7,3) ∉ R, because 7 is not less than 3.
Properties of Relations
- Reflexive: every element is related to itself. Example: "≤" relation, because x ≤ x is always true.
- Symmetric: if x is related to y, then y is related to x. Example: "=".
- Transitive: if x is related to y, and y to z, then x is related to z. Example: "<".
- Antisymmetric: if x is related to y and y to x, then x = y. Example: "≤".
Relations in Set Theory
Formally, a relation on a set X is a subset of the Cartesian product X × X. This means the relation consists of ordered pairs that indicate how the elements are related.
Summary
Relations describe connections between objects. Important properties include reflexivity, symmetry, transitivity, and antisymmetry.
Practice Exercise
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