Solutions To MAT206 15/16 Session Examination[Real Analysis Part I]

 

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MAT206 2015/16 Past Question
Check out my post on "Introduction to the real number system[Algebraic structures]" for better understanding.

(1a) Let $\mathbb{C}$  denote the system of complex numbers i.e ordered pairs $(x,y)$ of real numbers with the operations defined by
$(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)$
$(x_1,y_1)(x_2,y_2)=(x_1x_2-y_1y_2,x_1y_2+y_1x_2)$
Show that $\mathbb{C}$ is a filed?
(1b) Let $\mathbb{F}_3$ consists of 3 distinct elements $\theta,e,t$ where addition and multiplication is defined by  the table:

Solutions

(1a)      We are to show that $\mathbb{C}$ is a field, to do this, we show that $\mathbb{C}$ satisfies all axioms of field structure.
Now since $\mathbb{C}$ is an ordered pair $(x,y)$ of real numbers then the following field axioms hold:
If $x=(x_1,x_2)$ and $y=(y_1,y_2)$
(i) cummutativity $x+y=y+x$
(ii) additive $(x+y)=(x_1+y_1,x_2+y_2)$
(iii) Existence of a unique element $0$ such that
$0+x=x$
(iv)Existence of additive inverse: for every $x\in\mathbb{C}$ there exists a corresponding element $-x\in{\mathbb{C}}$ such that $x+(-x)=0$
Now we move onto the multiplicative property
(v) complex multiplication exists
$x.y=(x_1y_1-x_2y_2,x_1y_2+x_2y_1)$
(vi) multiplicative cummutativity
$x.y=y.x$
(vii) Existence multiplicative inverse if $x\in{\mathbb{C}}$ and $x\neq{0}$ then there exists a $\frac{1}{x}\in{\mathbb{C}}$ such that $x.(\frac{1}{x})=1$
(viii) Existence of multiplicative identity: $\mathbb{C}$ contains an element $1\neq{0}$ such that $1x=x$ for $x\in{C}$
(ix)Multiplicative distributivity
$x(y+z)=xy+xz$ for any pair $(x,y,z)\in{\mathbb{R}}$

(1b)     Let $\mathbb{F}_3$ consists of $3$ distinct element $\theta,,e,t$ where addition and multiplication is defined:

We now show that $\mathbb{F}_3$ is field using the given table
(i) Cummutativity $\theta+e=e+\theta=e$
(ii) Associativity $(\theta+e)+t=\theta+(e+t)=\theta$
(iii) Existence of additive unique element $0\in{\mathbb{F}_3}$ such that $0+\theta=\theta$
(iv) Existence of an inverse element $-\theta$ for any $\theta$
such that $\theta+(-\theta)=0$

Now we show for the multiplicative axioms

(v) Cummutativity $\theta.t=t.\theta=\theta$
(vi) Associativity $(\theta{e})t=\theta(et)=0$
(viii) Existence of a multiplicative inverse for any $\theta$ there exists $\theta^{-1}$ such that $\theta.\theta^{-1}=1$
(ix) Distributivity $\theta(e+t)=\theta{e}+\theta{t}=\theta$