📐 Formula Sheet — Logic Design

23 formulas across 3 chapters

Part 2: Introduction to Logic Design

$$V = \sum_{i=0}^{n} d_i \times r^i$$

Positional value formula: each digit $d_i$ is multiplied by $r^i$ where $r$ is the base and $i$ is the position (starting from 0 at the right).

$$\text{Decimal} = d_n \times r^n + d_{n-1} \times r^{n-1} + \cdots + d_0 \times r^0$$

Any base to decimal conversion using positional notation

$$\text{1's complement of } N = (2^n - 1) - N$$

Also: just flip every bit in the binary number

$$\text{2's complement of } N = 2^n - N = \text{1's complement} + 1$$

Used for subtraction: $A - B = A + (\text{2's complement of } B)$

$$G_i = B_i \oplus B_{i+1}, \quad G_{MSB} = B_{MSB}$$

Binary to Gray code conversion (XOR adjacent bits)

$$\text{Excess-3} = \text{BCD} + 0011$$

Excess-3 code = BCD representation + 3

Part 3: Boolean Algebra and Logic Gates

$$F = A \cdot B$$

AND operation (also written as $AB$)

$$F = A + B$$

OR operation

$$F = \overline{A} = A'$$

NOT (complement) operation

$$\text{NAND: } F = \overline{AB} = \overline{A} + \overline{B}$$

NAND = complement of AND (by De Morgan's)

$$\text{NOR: } F = \overline{A+B} = \overline{A} \cdot \overline{B}$$

NOR = complement of OR (by De Morgan's)

$$A \oplus B = A\overline{B} + \overline{A}B$$

XOR expanded as sum-of-products

$$\overline{A \oplus B} = AB + \overline{A}\,\overline{B}$$

XNOR expanded as sum-of-products

$$\overline{A \cdot B} = \overline{A} + \overline{B}$$

De Morgan's Theorem (AND form)

$$\overline{A + B} = \overline{A} \cdot \overline{B}$$

De Morgan's Theorem (OR form)

$$A + AB = A$$

Absorption Law

$$A + BC = (A+B)(A+C)$$

Distributive Law (OR over AND) — unique to Boolean algebra!

$$AB + \overline{A}C + BC = AB + \overline{A}C$$

Consensus Theorem — the BC term is redundant

Part 4: Minterms and Maxterms

$$F = \sum m(i, j, k, ...)$$

Canonical SOP notation: F equals the sum (OR) of minterms at positions i, j, k, ... (where F=1)

$$F = \prod M(i, j, k, ...)$$

Canonical POS notation: F equals the product (AND) of maxterms at positions i, j, k, ... (where F=0)

$$M_i = \overline{m_i}$$

A maxterm is the complement of the corresponding minterm

$$F = \sum m(i,j,...) \iff F = \prod M(\text{all indices not in } \{i,j,...\})$$

Converting canonical SOP ↔ POS: complement the index set

$$\overline{F} = \sum m(\text{maxterm indices of } F) = \prod M(\text{minterm indices of } F)$$

The complement of F uses the opposite index sets