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@@ -68,3 +68,97 @@ Please see [ECE 108: Discrete Math 1#Operator laws](/1b/ece108/#operator-laws) f
 The **synthesis** of an algebraic formula represents its implementation via logic gates. In this course, its total cost is the sum of all inputs to all gates and the number of gates, *excluding* initial inputs of "true" or an initial negation.
 
 In order to deduce an algebraic expression from a truth table, **OR** all of the rows in which the function returns true and simplify.
+
+??? example
+    Prove that $(x+y)\cdot(x+y')=x$:
+    
+    \begin{align*}
+    \tag{distributive property}(x+y)\cdot(x+y')&=xx+xy'+yx+yy' \\
+    \tag{$yy'$ = 0, $xx=x$}&=x + xy' + yx \\
+    \tag{distributive, commutative properties}&= x(1+y'+y) \\
+    \tag{1 + ... = 1}&= x(1) \\
+    &=x
+    \end{align*}
+    
+    Prove that $xy+yz+x'z=xy+x'z$:
+    
+    \begin{align*}
+    \tag{$x+x'=1$}xy+yz+x'z&=xy+yz(x+x')+x'z \\
+    \tag{distributive property}&=xy+xyz+x'yz+x'z \\
+    \tag{distributive property}&=x(y+yz) + x'(yz+z) \\
+    \tag{distributive property}&=xy(1+z) + x'z(y+1) \\
+    \tag{$1+k=1$}&=xy(1) + x'z(1) \\
+    \tag{$1\cdot k=k$}&= xy+x'z
+    \end{align*}
+
+### Minterms and maxterms
+
+The **minterm** $m$ is a **product** term where all variables in the function appear once. There are $2^n$ minterms for each function, where $n$ is the number of input variables.
+
+To determine the relevant function, the subscript can be converted to binary and each function variable set such that:
+
+- if the digit is $1$, the complement is used, and
+- if the digit is $0$, the original is used.
+
+$$m_j=x_1+x_2+\dots x_n$$
+
+!!! example
+    For a function that accepts three variables:
+    
+    - there are eight minterms, from $m_0$ to $m_7$.
+    - the sixth minterm $m_6=xyz'$ because $6=0b110$.
+    
+    For a sample function defined by the following minterms:
+    
+    $$
+    \begin{align*}
+    f(x_1,x_2,x_3)&=\sum m(1,2,5) \\
+    &=m_1+m_2+m_5 \\
+    &=x_1x_2x_3' + x_1x_2'x_3 + x_1'x_2x_3'
+    \end{align*}
+    $$
+
+The **maxterm** $M$ is a **sum** term where all variables in the function appear once. It is more or less the same as a minterm, except the condition for each variable is **reversed** (i.e., $0$ indicates the complement).
+
+$$M_j=x_1+x_2+\dots +x_n$$
+
+!!! example
+    For a sample function defined by the following maxterms:
+    
+    \begin{align*}
+    f(x_1,x_2,x_3,x_4)&=\prod M(1,2,8,12) \\
+    &=M_1M_2M_8M_{12} \\
+    \end{align*}
+
+??? example
+    Prove that $\sum m(1,2,3,4,5,6,7)=x_1+x_2+x_3$: **(some shortcuts taken for visual clarity)**
+    
+    \begin{align*}
+    \sum m(1,2,3,4,5,6,7) &=001+011+111+010+110+100+000 \\
+    \tag{SIMD distribution}&=001+010+100 \\
+    &=x_1+x_2+x_3
+    \end{align*}
+
+## VHDL
+
+VHDL is a hardware schematic language.
+
+<img src="https://static.javatpoint.com/tutorial/digital-electronics/images/multiplexer3.png" width=600 />
+
+For example, the basic 2-to-1 multiplexer expressed above can be programmed as:
+
+```vhdl
+entity two_one_mux is
+    port (a0, s, a1 : in    bit;
+          f         : out   bit);
+end two_one_mux
+
+architecture LogicFunc of two_one_mux is
+    begin
+        y <= (a0 AND s) OR (NOT s AND a1);
+    end LogicFunc;
+```
+
+In this case, the inputs are `a0, s, a1` that lead to an output `y`. All input/output is of type `bit` (a boolean).
+
+The **architecture** defines how inputs translate to outputs via functions. These all run **concurrently**.