Quick-reference cheatsheet for Boolean / digital-logic operator notation across textbooks, math, and Verilog.
| Symbol | Op | Reads as | Example |
|---|---|---|---|
| · ∧ & && * | AND | "and" / conjunction | $x \cdot y$, $x \wedge y$ |
| + ∨ | || | OR | "or" / disjunction | $x + y$, $x \vee y$ |
| ¬ ~ ! ′ ‾ | NOT | "not" / complement | $\overline{x}$, $\neg x$, $x'$ |
| Symbol | Op | Reads as | Equivalent |
|---|---|---|---|
| ⊕ ⊻ ^ | XOR | "exclusive or" | $x \oplus y = x\overline{y} + \overline{x}y$ |
| ⊙ ≡ ↔ | XNOR | "equivalence" | $x \odot y = \overline{x \oplus y}$ |
| ⊼ ↑ | NAND | "Sheffer stroke" | $x \uparrow y = \overline{x \cdot y}$ |
| ⊽ ↓ | NOR | "Pierce arrow" | $x \downarrow y = \overline{x + y}$ |
| Symbol | Op | Reads as / Meaning |
|---|---|---|
| → ⊃ ⇒ | implies | "if … then" — $x \to y = \overline{x} + y$ |
| ↔ ≡ | iff | "if and only if" (same as XNOR) |
| ∩ | intersection | set AND — same effect as · for sets |
| ∪ | union | set OR — same effect as + for sets |
| ⊂ ⊆ | subset | "is implied by" — rare in digital logic |
| Symbol | Meaning |
|---|---|
| 0 F ⊥ | false / logic 0 / GND |
| 1 T ⊤ | true / logic 1 / Vcc |
| ∀ | "for all" (used in proofs / SAT) |
| ∃ | "there exists" |
| Verilog | Op | Notes |
|---|---|---|
| & | AND | bitwise; &a = AND-reduction |
| | | OR | bitwise; |a = OR-reduction |
| ~ | NOT | bitwise complement |
| ^ | XOR | ^a = XOR-reduction (parity) |
| ~^ , ^~ | XNOR | bitwise equivalence |
| ~& | NAND | reduction NAND |
| ~| | NOR | reduction NOR |
| && || ! | logical | 1-bit boolean (returns 0 or 1) |
| == === | equality | logical / case (matches X, Z) |
| Symbol | Meaning |
|---|---|
| Σ m(…) | sum of minterms (canonical SOP) |
| Π M(…) | product of maxterms (canonical POS) |
| D(…) d(…) | don't-care set |
| $m_i$ | minterm i (product term that is 1 only on row i) |
| $M_i$ | maxterm i (sum term that is 0 only on row i) |
· for AND and + for OR, mirroring algebraic notation.∧ / ∨ / ¬.& / | / ^ perform the same operations bit-by-bit.