I have two boolean functions:
$$f(a, b, c, d) = \bar{a}cd + a\bar{c}d + ab\bar{d} + abc$$ and $$g(a, b, c, d) = d(a \oplus c) + ab$$
I know these two functions are equivalent because I tried every combination by using a truth table.
Now, can somebody help me show the equivalence of these two expressions by using basic boolean algebra laws?
$\def\onot#1{\overline{#1}}$Start with $g$, use the definition for $\oplus$ (exclusive or), and tautologies $a+\onot a$. Then use distribution, commutation, association, and idemptence.
$$\begin{split} g(a,b,c,d)&= d(a\oplus c)+ab \\ &=(\onot ac+a\onot c)(b+\onot b)d+ ab(c+\onot c)(d+\onot d)\\&~~\vdots\\ & =\onot acd+a\onot cd+ab\onot d+abc\\ & = f(a,b,c,d)\end{split}$$