# Lecture - Introduction to University Mathematics, Lecture 4

> Source: https://ollybritton.com/notes/uni/prelims/mt22/introduction-to-uni-maths/lectures/lecture-4/ · Updated: 2022-10-16 · Tags: uni, lecture

### Flashcards
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 $\le$?::
- Reflexive
- Anti-symmetric
- Transitive

What are the conditions for a relation $R$ to be a partial order relation?::
- Reflexive
- Anti-symmetric
- Transitive

What are the conditions for a relation $R$ to be an equivalence relation?::
- Reflexive
- Symmetric
- Transitive

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.

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