⚡ Cheatsheet: Digital Logic

Topic	Formula / Concept	Note
Boolean Algebra	\(A + \bar{A}B = A + B\)	Absorbption
	\(AB + \bar{A}C + BC = AB + \bar{A}C\)	Consensus Theorem
	\(f(A,B) = A \cdot f(1,B) + \bar{A} \cdot f(0,B)\)	Shannon Expansion
XOR Properties	\(A \oplus A = 0\), \(A \oplus 1 = \bar{A}\)	Odd function
	\(A \oplus B = \bar{A}B + A\bar{B}\)	Inequality detector
Logic Gates	NAND/NOR are Universal	Can implement any function
	\(\{ \oplus, 1 \}\) is NOT functional complete	Cannot implement AND/OR
Number Rep	Range of \(n\)-bit 2's Comp: \([-2^{n-1}, 2^{n-1}-1]\)	Asymmetric range
	Float Value: \((-1)^S \times 1.M \times 2^{E-Bias}\)	Bias = 127 (Single), 1023 (Double)
Mux	Formula to build \(N \times 1\) using \(M \times 1\): \(\lceil \frac{N-1}{M-1} \rceil\)	Useful for cascading
Counters	Mod-\(N\) counter freq: \(f_{out} = f_{clk} / N\)	Frequency division
	Ring Counter states: \(N\)	Johnson Counter states: \(2N\)
Delays	Ripple Adder Delay: \(N \times t_{FA}\)	Very slow for large \(N\)
	Lookahead Adder Delay: Constant (\(O(1)\) theoretically)	Larger area cost

⚠️ Common Traps

• minterm vs Maxterm: \(\sum m(1, 3)\) is equivalent to \(\Pi M(0, 2)\). Don't mix them up.
• 2's Complement: To negate a number, flip all bits and add 1. Negating \(0\) gives \(0\). Negating \(-2^{n-1}\) results in overflow.
• Overflow: In 2's comp addition, overflow occurs if adding two positives gives negative, or two negatives gives positive. \(V = C_{in} \oplus C_{out}\) (for MSB).