R-AESPE-2

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Sets of Numbers

First section of Chapter 6 in Cambridge (pg 44-42?)

There are many proofs

Direct proofs
Proof by contradiction
Proof by exhaustion
Proof by contrapositive
Proof by induction

Sets of numbers have symbols

$\mathbb{N}$: the set of natural numbers {1,2,3,4,5... $\infty$}
$\mathbb{Z}$: the set of all integers {$-\infty, \dots, -5,-4,-3,-2,-1,0, 1,2,3,4,5, \dots, \infty$}
$\mathbb{Q}$: the set of all rational numbers (numbers that can be written as $\frac{a}{b}$ where a and b are both integers)
$\mathbb{R}$: the set of all real numbers (includes all rational and irrational numbers)

See Number Sets Notes

Set notation

$\in$: means "is in" or "is a member of" or "belongs to".
- e.g. $x \in \mathbb{R}$ this means x is a member of, or is, a real number. It can be 3, it could be $\pi$.
$\subset$: means "is a subset of".
- e.g. $\mathbb{Z} \subset \mathbb{R}$, i.e. all integers are real numbesr.
$\subseteq$: for a subset where it doesnt completely contain the thing it is a subset of.

Conditional Statement

If a polygon is a quadrilaterial then it has exactly four sides.
- Stucture makes this a conditional statement, e.g. if -> then
- If Stated condition, then consequence
- Doesn't have to be the same structure
A conditional statement like "If A then B" can be written in math as:
- $\mathbb{A} \implies \mathbb{B}$

Proving Conditional Statement

When proving such a statement ($A \implies B$) you are proving a casual relationship exists between the truth o f A and the truth of B.

One can do this by:

Assume A is true (entertain hypothetical possibility), then show B must follow to be true as a consequence.

e.g. Prove If $n$ is divisible by 7, then $n^2$ is divisible by 49.

Proof

Assume that $n$ is divisible by 7. Thus $n=7m$ where $m \in \mathbb{Z}$. So $n^{2}= (7m)^{2} = 7^{2}m^{2} = 49m^{2}$ Since $m^2$ is an integer, $n^2$ is thus divisible by 49 QED (dont forget this lol)

Odd + Even Integers

When working with even integers, write as $2k, k \in \mathbb{Z}$

When working with odd integers, write as $2k + 1, k \in \mathbb{Z}$

If you need to show that a number is even, then you should show that it can be written as $2k, k \in \mathbb{Z}$

If you need to show that a number is odd, then you should show that it can be written as $2k + 1, k \in \mathbb{Z}$.