Question

Find the indicated set if$A=\left\{,2,3,4,5,6,7\right\}\text{}$,$B=\left\{2,4,6,8\right\}\text{}$and $C=\left\{7,8,9,10\right\}$

(a) $A\cup B\cup C$

(b)$A\cap B\cap C$

Short answer

(a) Union of $A,B$and $C$is given by $\text{A}\cup \text{B}\cup \text{C}=\left\{1,2,3,4,5,6,7,8,9,10\right\}$

(b) Intersection of $A,B$and $C$is given by $\text{A}\cap \text{B}\cap \text{C}=\varnothing$

Step-by-step solution

Part a. Step 1. Apply the concept of the union of sets.

If $S\mathrm{and}T$are sets, then their union $S\cup T$is the set that consists of all elements that are in$S\mathrm{or}T$ (or in both).

Part a. Step 2. Evaluate the given sets.

Given:

$\begin{array}{l}A=\left\{,2,3,4,5,6,7\right\}\text{}\\ B=\left\{2,4,6,8\right\}\text{}\\ C=\left\{7,8,9,10\right\}\end{array}$

1,2,3,4,5,6,7,8,9,10 are the numbers that are in $A\mathrm{or}B\mathrm{or}C$(or in all of them)

Part a. Step 3. Calculate the union of sets.

The union of $A,B$ and $C$ will be given by all the numbers that are in $A$ or Bor $C$(or in all of them)

$\begin{array}{l}\text{A}\cup \text{B}\cup \text{C}=\left\{1,2,3,4,5,6,7\right\}\cup \left\{2,4,6,8\right\}\cup \left\{7,8,9,10\right\}\\ \Rightarrow \text{A}\cup \text{B}\cup \text{C}=\left\{1,2,3,4,5,6,7,8,9,10\right\}\end{array}$

Part b. Step 1. Apply the concept of the intersection of sets.

The intersection of $S\mathrm{and}T$is the set $S\cap T$ consisting of all elements that are in both $S\mathrm{and}T$.

Part b. Step 2. Evaluate the given sets.

Given:

$\begin{array}{l}A=\left\{,2,3,4,5,6,7\right\}\text{}\\ B=\left\{2,4,6,8\right\}\text{}\\ C=\left\{7,8,9,10\right\}\end{array}$

There is no number that is common in all $A,B$and $C$.

Part b. Step 3. Calculate the intersection of sets.

The intersection of $A,B$ and $C$ will be given by the numbers that are common in all $A,B$and$C$

$\begin{array}{l}\text{A}\cap \text{B}\cap \text{C}=\left\{1,2,3,4,5,6,7\right\}\cap \left\{2,4,6,8\right\}\cap \left\{7,8,9,10\right\}\\ \Rightarrow \text{A}\cap \text{B}\cap \text{C}=\varnothing \end{array}$