From d95532688cef1b12922d3f6e51eb27dc1d00f634 Mon Sep 17 00:00:00 2001
From: lshprung <lshprung@yahoo.com>
Date: Fri, 25 Sep 2020 11:14:46 -0700
Subject: Post-class 09/25

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+[\<- Truth tables, minterms, and simple synthesis](2.md)
+
+---
+
+# Boolean Algebra and DeMorgan's Theorem
+
+## Boolean algebra basics
+
+### Boolean Algebra Properties
+
+- Commutative (input order doesn't matter)
+	- `X*Y` = `Y*X`
+	- `X+Y` = `Y+X`
+- Associative (can combine like functions)
+	- `X*(Y*Z)` = `(X*Y)*Z` = `X*Y*Z`
+	- `X+(Y+Z)` = `(X+Y)+Z` = `X+Y+Z`
+- Distributive (going the other way: factoring)
+	- `X * (Y+Z)` = `X*Y + X*Z`
+- Combining
+	- `X * Y + X * !Y` = `X * (Y + !Y)` = `X`
+
+---
+
+## DeMorgan's Theorem
+
+- `!(X*Y)` = `!X + !Y` (and `!(X+Y)` = `!X * !Y`)
+- Allows us to transform any AND gate to an OR gate, or vice versa
+- NOTE: `!(X*Y)` != `!X * !Y`
+
+|x|y| |x\*y|!(x\*y)| |!x|!y|!x + !y|
+|-|-|-|----|-------|-|--|--|-------|
+|0|0| |0   |1      | |1 |1 |1      |
+|0|1| |0   |1      | |1 |0 |1      |
+|1|0| |0   |1      | |0 |1 |1      |
+|1|1| |1   |0      | |0 |0 |0      |
+
+### NAND and NOR gates
+
+- An AND gate with an inverter on the output is called a NAND gate
+- An OR gate with an inverter on the output is called a NOR gate
+
+![diagram](3.1.png)
+
+### Visualizing DeMorgan
+
+- "Flip" the gate type (AND \<-> OR), invert all the inputs, invert the output
+	- Remember that bubbles mean inverters
+	- Can be applied to any gate/inverter combo
+
+![diagram](3.2.png)
+
+### A note about inversions
+
+- To be clear: inverting an inversion essentially cancels it out
+	- If you invert a value and then invert it again, you're back to where you started
+- When applying DeMorgan's, inverting an inversion makes it go away
+
+![diagram](3.3.png)
+
+### Applying DeMorgan
+
+- Inversions can be added in pairs
+
+![diagram](3.4.png)
+
+- Inversions can "slide" to the other end of a wire
+
+![diagram](3.5.png)
+
+---
+
+## Other Synthesis Approaches: solving for 0's, POS, reducing with Boolean Algebra
+
+- Say we had f(x1,x2,x3) = ∑m(1,4,5,6)
+	- What does the truth table look like?
+	- What is the canonical SOP equation?
+	- What does the matching circuit look like?
+- What are some other ways we can synthesize this function?
+	- Synthesize the inverse function, then invert
+		- Simpler circuit if more 1's than 0's in truth table (which is actually not the case here)
+	- Reduce the equation using Boolean algebra
+
+### Truth table for inverse function
+
+- If F(x1,x2,x3) = ∑m(1,4,5,6), then !F = ∑m(0,2,3,7)
+
+|x1x2x3|F|!F|
+|------|-|--|
+|000   |0|1 |
+|001   |1|0 |
+|010   |0|1 |
+|011   |0|1 |
+|100   |1|0 |
+|101   |1|0 |
+|110   |1|0 |
+|111   |0|1 |
+
+### Synthesizing the inverse
+
+- Create equation for !f by adding minterms that evaluate to 0
+	- !f = m0 + m2 + m3 + m7
+	- f = !(m0 + m2 + m3 + m7) (a valid answer)
+- A step further would apply deMorgan's 
+	- f = !m0 * !m2 * !m3 * !m7
+- One step further would use maxterms
+	- ex. !m0 = M0 = !(!x1 * !x2 * !x3) = x1 + x2 + x3
+	- f = M0 * M2 * M3 * M7 (yet another valid answer)
+	- Called product of sums (POS) form
+
+### Another POS example
+
+- Say F(A,B,C) = ∑m(0,2,3,4,6,7)
+	- !F = ∑m(1,5)
+
+![diagram](3.6.png)
+
+### Reduction w/ Boolean Algebra
+
+- f = `(!x1 * !x2 * x3) + (x1 * !x2 * !x3) + (x1 * !x2 * x3) +(x1 * x2 * !x3)` (canonical SOP)
+- f = `((!x1 + x1) * !x2 * x3) + (x1 * (!x2 + x2) * !x3)`
+- f = `(!x2 * x3) + (x1 * !x3)`
+- The best answer yet, but it can be hard to see these reductions in equation form
+
+![diagram](3.7.png)
-- 
cgit