Basic logic symbols
|
Symbol
|
Name | Explanation | Examples | Unicode Value |
HTML Name |
LaTeX symbol |
|---|---|---|---|---|---|---|
| Read as | ||||||
| Category | ||||||
|
⇒
→ ⊃ |
material implication | A ⇒ B is true only in the case that either A is false or B is true.
→ may mean the same as ⇒ (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols). ⊃ may mean the same as ⇒ (the symbol may also mean superset). |
x = 2 ⇒ x2 = 4 is true, but x2 = 4 ⇒ x = 2 is in general false (since x could be −2). | U+21D2
U+2192 U+2283 |
⇒
→ ⊃ |
 \Rightarrow \to \supset \implies |
| implies; if .. then | ||||||
| propositional logic, Heyting algebra | ||||||
|
⇔
≡ ↔ |
material equivalence | A ⇔ B is true only if both A and B are false, or both A and B are true. | x + 5 = y + 2 ⇔ x + 3 = y | U+21D4
U+2261 U+2194 |
⇔
≡ ↔ |
 \Leftrightarrow \equiv \leftrightarrow \iff |
| if and only if; iff; means the same as | ||||||
| propositional logic | ||||||
|
¬
˜ ! |
negation | The statement ¬A is true if and only if A is false.
A slash placed through another operator is the same as „¬” placed in front. |
¬(¬A) ⇔ A x ≠ y ⇔ ¬(x = y) |
U+00AC
U+02DC |
¬
˜ ~ |
 \lnot or \neg \sim |
| not | ||||||
| propositional logic | ||||||
|
∧
• & |
logical conjunction | The statement A ∧ B is true if A and B are both true; else it is false. | n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number. | U+2227
U+0026 |
∧
& |
 \wedge or \land\&[2] |
| and | ||||||
| propositional logic, Boolean algebra | ||||||
|
∨
+ ǀǀ |
logical (inclusive) disjunction | The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false. | n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number. | U+2228 | ∨ |  \lor or \vee |
| or | ||||||
| propositional logic, Boolean algebra | ||||||
|
⊕
⊻ |
exclusive disjunction | The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same. | (¬A) ⊕ A is always true, A ⊕ A is always false. | U+2295
U+22BB |
⊕ |  \oplus \veebar |
| xor | ||||||
| propositional logic, Boolean algebra | ||||||
|
⊤
T 1 |
Tautology | The statement ⊤ is unconditionally true. | A ⇒ ⊤ is always true. | U+22A4 | T |  \top |
| top, verum | ||||||
| propositional logic, Boolean algebra | ||||||
|
⊥
F 0 |
Contradiction | The statement ⊥ is unconditionally false. | ⊥ ⇒ A is always true. | U+22A5 | ⊥ F |  \bot |
| bottom, falsum, falsity | ||||||
| propositional logic, Boolean algebra | ||||||
|
∀
() |
universal quantification | ∀ x: P(x) or (x) P(x) means P(x) is true for all x. | ∀ n ∈ ℕ: n2 ≥ n. | U+2200 | ∀ |  \forall |
| for all; for any; for each | ||||||
| first-order logic | ||||||
|
∃
|
existential quantification | ∃ x: P(x) means there is at least one x such that P(x) is true. | ∃ n ∈ ℕ: n is even. | U+2203 | ∃ |  \exists |
| there exists | ||||||
| first-order logic | ||||||
|
∃!
|
uniqueness quantification | ∃! x: P(x) means there is exactly one x such that P(x) is true. | ∃! n ∈ ℕ: n + 5 = 2n. | U+2203 U+0021 | ∃ ! |  \exists ! |
| there exists exactly one | ||||||
| first-order logic | ||||||
|
:=
≡ :⇔ |
definition | x := y or x ≡ y means x is defined to be another name for y (but note that ≡ can also mean other things, such as congruence).
P :⇔ Q means P is defined to be logically equivalent to Q. |
cosh x := (1/2)(exp x + exp (−x))
A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B) |
U+2254 (U+003A U+003D)
U+2261 U+003A U+229C |
:= : ≡ ⇔ |
 :=\equiv\Leftrightarrow |
| is defined as | ||||||
| everywhere | ||||||
|
( )
|
precedence grouping | Perform the operations inside the parentheses first. | (8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1, but 8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4. | U+0028 U+0029 | ( ) |  ( ) |
| parentheses, brackets | ||||||
| everywhere | ||||||
|
⊢
|
Turnstile | x ⊢ y means y is provable from x (in some specified formal system). | A → B ⊢ ¬B → ¬A | U+22A2 | ⊢ |  \vdash |
| provable | ||||||
| propositional logic, first-order logic | ||||||
|
⊨
|
double turnstile | x ⊨ y means x semantically entails y | A → B ⊨ ¬B → ¬A | U+22A8 | ⊨ |  \vDash |