Chapter 6.BMO, Problem 1BMO

To determine

a)

To find:

The set of $A\cap \left(B\cup C\right)$.

Answer to Problem 1BMO

Solution:

The set of $A\cap \left(B\cup C\right)$ is $\left\{d,f,g\right\}$.

Explanation of Solution

Given:

The elements of a universal set $U$ are $\left\{a,b,c,d,e,f,g\right\}$, the elements of the set $A$ are $\left\{a,d,f,g\right\}$, the elements of the set $B$ are $\left\{d,f,g\right\}$ and the elements of the set $C$ are $\left\{b,c,e,f\right\}$.

Calculation:

Write the equation of the given sets.

$A\cap \left(B\cup C\right)$

Substitute $\left\{a,d,f,g\right\}$ for A, $\left\{d,f,g\right\}$ for B and $\left\{b,c,e,f\right\}$ for $C$ in the above expression.

$\begin{array}{rll}A\cap \left(B\cup C\right)& =& \left\{a,d,f,g\right\}\cap \left(\left\{d,f,g\right\}\cup \left\{b,c,e,f\right\}\right)\\ & =& \left\{a,d,f,g\right\}\cap \left\{b,c,d,e,f,g\right\}\\ & =& \left\{d,f,g\right\}\end{array}$

Therefore, the set of $A\cap \left(B\cup C\right)$ is $\left\{d,f,g\right\}$.

Conclusion:

Hence, the set of $A\cap \left(B\cup C\right)$ is $\left\{d,f,g\right\}$.

To determine

b)

To find:

The set of $\left(A\cap C\right)\cup \left(B\cup C\right)$.

Answer to Problem 1BMO

Solution:

The set of $\left(A\cap C\right)\cup \left(B\cup C\right)$ is $\left\{b,c,d,e,f,g\right\}$.

Explanation of Solution

Calculation:

Write the equation of the given sets.

$\left(A\cap C\right)\cup \left(B\cup C\right)$

Substitute $\left\{a,d,f,g\right\}$ for A, $\left\{d,f,g\right\}$ for B and $\left\{b,c,e,f\right\}$ for $C$ in the above expression.

$\begin{array}{rll}\left(A\cap C\right)\cup \left(B\cup C\right)& =& \left(\left\{a,d,f,g\right\}\cap \left\{b,c,e,f\right\}\right)\cup \left(\left\{d,f,g\right\}\cup \left\{b,c,e,f\right\}\right)\\ & =& \left\{f\right\}\cup \left\{b,c,d,e,f,g\right\}\\ & =& \left\{b,c,d,e,f,g\right\}\end{array}$

Therefore, the set of $\left(A\cap C\right)\cup \left(B\cup C\right)$ is $\left\{b,c,d,e,f,g\right\}$.

Conclusion:

Hence, the set of $\left(A\cap C\right)\cup \left(B\cup C\right)$ is $\left\{b,c,d,e,f,g\right\}$.

To determine

c)

To find:

The set of $A^c$.

Answer to Problem 1BMO

Solution:

The set of $A^c$ is $\left\{b,c,e\right\}$.

Explanation of Solution

Approach:

The formula used to calculate the complement of $A$.

$A^c=U-A$ … (1)

Here, $U$ is the universal set and $A^c$ is the complement of $A$.

Calculation:

Substitute $\left\{a,d,f,g\right\}$ for A and $\left\{a,b,c,d,e,f,g\right\}$ for $U$ in the formula (1) to calculate the set of $A^c$.

$\begin{array}{rll}{A}^c& =& \left\{a,b,c,d,e,f,g\right\}-\left\{a,d,f,g\right\}\\ & =& \left\{b,c,e\right\}\end{array}$

Therefore, the set of $A^c$ is $\left\{b,c,e\right\}$.

Conclusion:

Hence, the set of $A^c$ is $\left\{b,c,e\right\}$.