## Exercises5.2Exercises

###### 2.

Prove the commutative property expressed by Equation (5.1.5) and Equation (5.1.6).

Solution

For Equation (5.1.5):

 $$x$$ $$y$$ $$x \cdot y$$ $$y \cdot x$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$

And for Equation (5.1.6):

 $$x$$ $$y$$ $$x + y$$ $$y + x$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$
###### 4.

Prove the complement property expressed by Equation (5.1.9) and Equation (5.1.10).

Solution

For Equation (5.1.9):

 $$x$$ $$x'$$ $$x \cdot x'$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{0}$$

And for Equation (5.1.10):

 $$x$$ $$x'$$ $$x + x'$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{1}$$
###### 5.

Prove the idempotent property expressed by Equation (5.1.11) and Equation (5.1.12).

Solution
 $$x$$ $$x$$ $$x \cdot x$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$

And for Equation (5.1.12):

 $$x$$ $$x$$ $$x + x$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$
###### 6.

Prove the distributive property expressed by Equation (5.1.13) and Equation (5.1.14).

Solution
 $$x$$ $$y$$ $$z$$ $$y + z$$ $$x \cdot (y + z)$$ $$x \cdot y$$ $$x \cdot z$$ $$x \cdot y + x \cdot z$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$

And for Equation (5.1.14):

 $$x$$ $$y$$ $$z$$ $$y \cdot z$$ $$x + y \cdot z$$ $$x + y$$ $$x + z$$ $$(x + y) \cdot (x + z)$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{0}$$ $$\binary{0}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$ $$\binary{1}$$