Question

List the elements of $\mathit{A}\mathbf{\times }\mathit{B}\mathbf{\times }\mathit{C}$when $\mathit{A}\mathbf{=}\left\{0,1\right\}$and $\mathit{B}\mathbf{,}\mathit{C}$ are as in Exercise 7. (Exercise 7: List the elements of $\mathit{B}\mathbf{\times }\mathit{C}$ when $\mathit{B}\mathbf{=}\left\{a,b,c\right\}$and $\mathit{C}\mathbf{=}\left\{0,1,c\right\}$.)

Short answer

The value of $A\times B\times C$is $\left\{\begin{array}{l}\left(0,a,0\right),\left(0,a,1\right),\left(0,a,c\right),\left(0,b,0\right),\left(0,b,1\right),\\ \left(0,b,c\right),\left(0,c,0\right),\left(0,c,1\right),\left(0,c,c\right),\\ \left(1,a,0\right),\left(1,a,1\right),\left(1,a,c\right),\left(1,b,0\right),\left(1,b,1\right),\\ \left(1,b,c\right),\left(1,c,0\right),\left(1,c,1\right),\left(1,c,c\right)\end{array}\right\}$.

Step-by-step solution

Consider the given statement

The sets are $\mathrm{A}=\left\{0,1\right\}$, $B=\left\{a,b,c\right\}$ and $C=\left\{0,1,c\right\}$.

List the elements of the sets A×B×C 

The value of $A\times B\times C$can be listed as follows:

$\begin{array}{c}A\times B\times C=\left\{0,1\right\}\times \left\{a,b,c\right\}\times \left\{0,1,c\right\}\\ =\left\{0,1\right\}\times \left\{\left(a,0\right),\left(a,1\right),\left(a,c\right),\left(b,0\right),\left(b,1\right),\left(b,c\right),\left(c,0\right),\left(c,1\right),\left(c,c\right)\right\}\\ =\left\{\begin{array}{l}\left(0,a,0\right),\left(0,a,1\right),\left(0,a,c\right),\left(0,b,0\right),\left(0,b,1\right),\\ \left(0,b,c\right),\left(0,c,0\right),\left(0,c,1\right),\left(0,c,c\right),\\ \left(1,a,0\right),\left(1,a,1\right),\left(1,a,c\right),\left(1,b,0\right),\left(1,b,1\right),\\ \left(1,b,c\right),\left(1,c,0\right),\left(1,c,1\right),\left(1,c,c\right)\end{array}\right\}\end{array}$

Therefore, the value of$A\times B\times C$ is $\left\{\begin{array}{l}\left(0,a,0\right),\left(0,a,1\right),\left(0,a,c\right),\left(0,b,0\right),\left(0,b,1\right),\\ \left(0,b,c\right),\left(0,c,0\right),\left(0,c,1\right),\left(0,c,c\right),\\ \left(1,a,0\right),\left(1,a,1\right),\left(1,a,c\right),\left(1,b,0\right),\left(1,b,1\right),\\ \left(1,b,c\right),\left(1,c,0\right),\left(1,c,1\right),\left(1,c,c\right)\end{array}\right\}$$\left\{\begin{array}{l}\left(0,a,0\right),\left(0,a,1\right),\left(0,a,c\right),\left(0,b,0\right),\left(0,b,1\right),\\ \left(0,b,c\right),\left(0,c,0\right),\left(0,c,1\right),\left(0,c,c\right),\\ \left(1,a,0\right),\left(1,a,1\right),\left(1,a,c\right),\left(1,b,0\right),\left(1,b,1\right),\\ \left(1,b,c\right),\left(1,c,0\right),\left(1,c,1\right),\left(1,c,c\right)\end{array}\right\}$.