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An Introduction to Basic Set Theory

Transitive and Antisymmetric Relations

Learn to identify transitive relations where if (a, b) and (b, c) are in a relation then (a, c) must be included, and antisymmetric relations where concurrent (a, b) and (b, a) imply a equals b. Discover how to verify these properties using Python code and examples to deepen your understanding of relation types in set theory.

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Transitive relation

A relation RR on a set AA is a transitive relation if for any two ordered pairs (a,b)∈R(a,b)\in R and (b,c)∈R(b,c)\in R, then the ordered pair (a,c)∈R(a,c) \in R. The relation RR is not transitive if there are ordered pairs (a,b)∈R(a,b)\in R and (b,c)∈R(b,c)\in R so that (a,c)∉R(a,c)\not\in R.

Let’s take the set M={α,β,γ}M= \{\alpha, \beta, \gamma\} to see some examples. The following relations on MM are examples of transitive relations:

Now, let’s look at the relation R8R_8 shown in the image below to see why it’s transitive. Because (α,β)∈R8(\alpha , \beta)\in R_8 and (β,γ)∈R8(\beta , \gamma)\in R_8, (α,γ)(\alpha, \gamma) should be in R8R_8, and we observe that (α,γ)∈R8(\alpha, \gamma)\in R_8. Similarly, because (α,β)∈R8(\alpha , \beta)\in R_8 ...