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Relations, functions and equivalence

Relations as sets of pairs; reflexive, symmetric, transitive; functions and bijections.

A relation is a set of ordered pairs; an equivalence relation (reflexive, symmetric, transitive) partitions its set into classes; a function is a relation with exactly one output per input. Picture it: "same remainder mod 3" splits the integers into three columns. Think it: a bijection is how we say two sets have the same size, including infinite ones.

Àwọn ààtò àwọn ìṣàmúlò-ètò · truth table of p implies q

Discrete-Math · Propositional-Logic

Truth table of p implies q

Àwọn ìṣàmúlò-ètò

p \Rightarrow q
2 variable(s) → 4 rows. Fill in every combination.
\
Contingent: true for some inputs, false for others.

Fi àwọn àgbèwọlé hàn

p \Rightarrow q

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Bii o ṣe le: Relations, functions and equivalence

2 variable(s) → 4 rows. Fill in every combination.
Contingent: true for some inputs, false for others.

Àwọn Àtòjọ-ẹ̀yàn

Are some infinities bigger than others?

Yes. The integers and the rationals can be listed; the real numbers cannot (Cantor's diagonal argument), so there are strictly more reals than integers.

What is the difference between a relation and a function?

A relation pairs inputs with outputs freely; a function is a relation in which every input gets exactly one output.

Diẹ̀ nínú Set Theory & Logic

Sets and operations Cardinality and infinity Logic and methods of proof