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Tutorial covering relation properties: antisymmetry, symmetry, and transitivity. Focus on proof techniques using counterexamples and working with formal definitions to determine whether relations satisfy these properties.

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Key definitions

Antisymmetric relation:

A relation R is antisymmetric if for all x, y:

• If (x, y) ∈ R and (y, x) ∈ R, then x = y

To show a relation is not antisymmetric, find one counterexample where:

• Both (x, y) and (y, x) are in R
• But x ≠ y

Symmetric relation:

A relation R is symmetric if for all x, y:

• If (x, y) ∈ R, then (y, x) ∈ R

To show a relation is not symmetric, find one counterexample where:

• (x, y) ∈ R
• But (y, x) ∉ R

Transitive relation:

A relation R is transitive if for all x, y, z:

• If (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R

Problem 1: Relation R with equality condition

Relation definition:

R = {((a,b), (c,d)) | a = c and b ≠ d}