|
Symbol
|
Name |
Explanation |
Examples |
Unicode Value |
| Should be read as |
| Category |
|
⇒
→
⊃
|
material implication |
A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B.
→ 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). |
8658
8594
8835 |
| implies; if .. then |
| propositional logic, Heyting algebra |
|
⇔
≡
↔
|
material equivalence |
A ⇔ B means A is true if B is true and A is false if B is false. |
x + 5 = y +2 ⇔ x + 3 = y |
8660
8596 |
| if and only if; iff |
| propositional logic |
|
¬
˜
|
logical 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) |
172
732 |
| 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. |
8743
38 |
| and |
| propositional logic |
|
∨
|
logical 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. |
8744 |
| or |
| propositional logic |
⊕
⊻
|
exclusive or |
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. |
8853
8891 |
| xor |
| propositional logic, Boolean algebra |
⊤
T
1
|
logical truth |
The statement ⊤ is unconditionally true. |
A ⇒ ⊤ is always true. |
8868 |
| top |
| propositional logic, Boolean algebra |
⊥
F
0
|
logical falsity |
The statement ⊥ is unconditionally false. |
⊥ ⇒ A is always true. |
8869 |
| bottom |
| propositional logic, Boolean algebra |
|
∀
|
universal quantification |
∀ x: P(x) means P(x) is true for all x. |
∀ n ∈ N: n2 ≥ n. |
8704 |
| for all; for any; for each |
| predicate logic |
|
∃
|
existential quantification |
∃ x: P(x) means there is at least one x such that P(x) is true. |
∃ n ∈ N: n is even. |
8707 |
| 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. |
8707 33 |
| 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) |
58 61
8801
58 8660 |
| 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. |
40 41 |
|
| everywhere |
|
⊢
|
inference |
x ⊢ y means y is derived from x. |
A → B ⊢ ¬B → ¬A |
8866 |
| infers or is derived from |
| propositional logic, first-order logic |