boolean algebra: simplify $ a* b *d + \tilde a *\tilde c*d + b* \tilde c* d$

Question

Simplify the following function(algebraically): $$y = a*b*d + \tilde a *\tilde c*d + b *\tilde c *d$$

the solution is: $$a*b*d + \tilde a * \tilde c * d$$ which i checked via karnaugh and also wolfram.

my "solution" so far: $b*d*(a + \tilde c) + (\tilde a * \tilde c * d)$ there should be a rule that should eliminate the $\tilde c$ inside the $(a +\tilde c)$ term. Does anybody know how to do it?

Answer 1

Hint: write $$b*\tilde c*d = a*b*\tilde c*d + \tilde a*b*\tilde c*d$$ and then see if you can absorb

• $a*b*\tilde c*d$ into $a*b*d$

• and $\tilde a*b*\tilde c*d$ into $\tilde a*\tilde c*d$

using $x+ x*y = x*(1+y) = x*1 = x$ for suitable choices of $x$ and $y$.