Week 1 Summary
Notion of Proof
Types of Numbers
- Natural numbers $\mathbb{N}$ are whole numbers that start from 0.
- Integers are whole numbers including negatives.
- Rational numbers are any number that can be represented as fractions without a denominator of 0.
- Real numbers are any decimal number that can be presented on a number line. E.g. $\pi$
Proofs
A mathematical proof is a carefully reasoned argument to convince a sceptical listener that a statement is true.
Properties of Odd and Even Numbers
An even number $m$ is in the form $m = 2k$ where $k$ is an integer.
An odd number $n$ is in the form $n= 2l+1$ where $l$ is an integer.
Notation
- The symbol $\exists$ means exists.
- The symbol $\Leftrightarrow$ means if and only if.
- The symbol $\forall$ means for all.
Some definitions written in notation may look like:
-
$n$ is even $\Leftrightarrow \exists$ an integer $k$ such that $n=2k$
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$n$ is odd $\Leftrightarrow \exists$ an integer $k$ such that $n=2k+1$