Question

If$\mathit{C}$ is a finite set, then$\mathbf{\left|}\mathbf{C}\mathbf{\right|}$ denotes the number of elements in$\mathit{C}$. If $\mathit{A}$and $\mathit{B}$are finite sets, is it true that$\mathbf{|}\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{|}\mathbf{=}\mathbf{\left|}\mathbf{A}\mathbf{\right|}\mathbf{+}\mathbf{\left|}\mathbf{B}\mathbf{\right|}$ .

Short answer

The required statement $|A\cup B|=\left|A\right|+\left|B\right|$is proved.

Step-by-step solution

Write the given data from the question.

The$C$ is the finite set, $\left|C\right|$denotes the number of elements in $C$.

The $A$and $B$are the finite sets.

Prove that|A∪B|=|A|+|B| .

It is given that$A$ and$B$ are the finite sets therefore $m\in N\cup \left\{0\right\}$.

The bijection function is expressed as,$f:A\to \left[k\right]$.

There is also$n\in N-\left\{0\right\}$ and bijection function expression as$g:B\to \left[j\right]$.

Let suppose the combined bijection function is represents by $h$and can be expressed as,$h:A\cup B\to [k+j]$.

$h\left(x\right)=\{\begin{array}{l}f\left(x\right)x\in A\\ k+g\left(x\right)x\in B\end{array}$

From above it is verified that $h$is bijection.

$\begin{array}{l}|A\cup B|=k+j\\ |A\cup B|=\left|A\right|+\left|B\right|\end{array}$

Hence the required statement$|A\cup B|=\left|A\right|+\left|B\right|$is proved.