Lecture - Introduction to University Mathematics, IV

What does the notation $\exists !$ mean?

“There exists unique”.

What does it mean for $a\, R\, b$ for a relation $R$ on sets $A$ and $B$?

$(a, b) \in A \times B$

What does it mean for a relation $R$ on $S$ to be reflexive?

\[\forall x : x\, R\, x\]

What does it mean for a relation $R$ on $S$ to be symmetric?

\[\forall x, y \in S, x\ R\ y \iff y \ R \ x\]

What does it mean for a relation $R$ on $S$ to be anti-symmetric?

\[\forall x, y \in S, x\ R\ y \land y \ R \ x \implies x = y\]

What does it mean for a relation $R$ on $S$ to be transitive?

\[x\ R\ y \land y\ R \ z \implies x\ R\ z\]

Which out of (reflexive, symmetric, anti-symmetric, transitive) is the relation $\le$?

What are the conditions for a relation $R$ to be a partial order relation?

What are the conditions for a relation $R$ to be an equivalence relation?

What’s the notation for $a$ is equivalent to $b$ under some equivalence relation?

\[a \sim b\]

What is the notation for the equivalence class of $x$?

\[\bar{x}\]

What is a partition of a set $S$?

A collection of non-empty disjoint subsets who’s union is $S$.

How can you relate equivalence relations and partitions on sets?

Say two objects are only equivalent if they are in the same part of a partition.