Logical Expression

Question

I saw this in the text.

F = ~D~H~L + D~H~L + D~HL

and it said this is equivalent to

F = D~H + ~D~H~L

Anyone know what is the process to this equivalence. I try to solved it but I'm not able to.

~ is "not"

Answer 1

For better readability I’ll write $\bar{D}$ for your $\sim D$. Then

$$\begin{align*} \bar{D}\bar{H}\bar{L}+D\bar{H}\bar{L}+D\bar{H}L&\equiv(\bar D\bar L+D\bar L+DL)\bar H\\ &\equiv\big(\bar D\bar L+D(\bar L+L)\big)\bar H\\ &\equiv(\bar D\bar L+D)\bar H\\ &\equiv\bar D\bar L\bar H+D\bar H\;. \end{align*}$$