## Example 1

Let $$t = \begin{bmatrix} 0 \\ 1 \\ 2\end{bmatrix}+ 2\begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}$$ be a $$3$$-tuple with entries from $$\mathbb{R}$$. What is $$t_2^2$$?

## Example 2

Write out all the elements of the set $$\{n \in \mathbb{N} : (n-2)^2 \leq 10\}$$.

## Example 3

Write out all the elements of the set $$\left\{\begin{bmatrix} a \\ b \end{bmatrix} \in \mathbb{Z}^2 : |a+b|=1,~a\geq 0,~b \geq -1\right\}$$.

## Example 4

Evaluate the expression $$1 + (1 + (1 + 1)\cdot 1)$$ over $$GF(2)$$.

## Example 5

Give the multiplicative inverse of $$4-\sqrt{5}$$ in the form $$a+b\sqrt{5}$$ where $$a,b \in \mathbb{Q}$$.

## Example 6

Let $$a \in \mathbb{F}$$ where $$\mathbb{F}$$ denotes a field. Suppose that $$a \neq 0$$. Show that $$a^{-k} = \left(a^{-1}\right)^k$$ for every natural number $$k$$. (You may use the fact that $$a^k b^k = (ab)^k$$ for all $$a,b \in \mathbb{F})$$.

## Example 7

Let $$\mathbb{Q}(\sqrt{2})$$ denote the set of numbers of the form $$a + b\sqrt{2}$$ where $$a$$ and $$b$$ are rational numbers. Show that $$\mathbb{Q}(\sqrt{2})$$ under the usual addition and multiplication for real numbers form a field.

## Example 8

Consider the function $$T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$$ given by $T\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} 2x - y \\ x + y \end{bmatrix}.$ Show that $$T$$ is surjective.