Given a proof, I tried to analyze it and identify the properties of boolean algebra used at each step. However, I am stumped with the first line: $(a+b) + \sim(a) \cdot \sim(b) = (a + b + \sim(a))(a+b+\sim(b))$. Which property makes those two things equal?
To show $\sim(a+b) = \sim(a) \cdot \sim(b)$
Proof
Show $(a+b) + \sim(a) \cdot \sim(b) = 1$ $$ \begin{align} (a+b) + \sim(a) \cdot \sim(b) &= (a + b + \sim(a))(a+b+\sim(b)) \\ &= (a+\sim(a) + b)(a + b+ \sim(b)) \\ &=(1+b)(a+1) \\ &=1\cdot 1 = 1 \\ \end{align} $$
Next, show $(a+b) \cdot \sim(a) \cdot \sim(b) = 0$
$$\begin{align} (a+b) \cdot \sim(a) \cdot \sim(b) &= a \cdot \sim(a) \cdot \sim(b) + \sim(a) \cdot \sim(b) \\ &= a \cdot \sim(a) \cdot \sim(b) + \sim(a) \cdot b \cdot \sim(b) \\ &=0 \cdot \sim(b) + \sim(a) \cdot 0 \\ &= 0 + 0 = 0 \end{align} $$
We conclude that $\sim(a) \cdot \sim(b)$ is the complement of $a+b$ (because those equations would not have worked otherwise).
To show $\sim(a\cdot b) = \sim(a) + \sim(b)$
... (to be completed once I understand the properties used above)
Properties Used in DeMorgan's Law:
Property of 0- X+0=X ; X⋅0=0
Property of 1- X+1=1 ; X⋅1=X
Idempotence Law- X+X=X ; X⋅X=X
Involution Law- (X')'=X
Complementarity Law - X+X'=1 ; X⋅X'=0
Commutative Law-X+Y=Y+X ; X⋅Y=Y⋅X
Associative Law- (X+Y)+Z=X+(Y+Z); (X⋅Y)⋅Z=X⋅(Y⋅Z)
Distributive Law-X(Y+Z)=XY+XZ ; X+YZ=(X+Y)(X+Z)
Here in these two links you will find the proper boolean algebra proof of DeMorgan's Law along with the property used in each statement:
http://mines.humanoriented.com/410/books/boolean_algebra.pdf
http://www2.nau.edu/~sh295/EE110/deMorganproof.html
Hope This Helped✌