Notation
This is the notation resource.
Operator: Its simplest meaning in mathematics and logic, is an action or procedure which produces a new value from one or more input values. There a unary, binary, and ternary operators.
Arithmetic Operations...
Operator / Operation | Notation | Syntax | Extra |
---|---|---|---|
Add, addition | + | A + B | |
Subtract, subtraction | - | A - B | |
Multiply, multiplication | * or x | A * B | A + A + A... |
Divide, division | ÷ or / | A ÷ B | A ÷ A ÷ A... |
Basic Relations...
Relation | Notation | Syntax | Extra |
---|---|---|---|
Equals | = | A = B | |
Not equals | ≠or != or <> | A≠B | |
Greater than | > | A > B | |
Less than | < | A < B | |
Greater than or equals | > or >= | A > B | |
Lesser than or equals | < or <= | A < B |
Set Theory Operations...
Operator / Operation | Notation | Syntax | Extra |
---|---|---|---|
Union | ∪ | A ∪ B | Or |
Intersection | ∩ | A ∩ B | And |
Complement | ^{c} | A^{c} | Set difference |
Cartesian Product | × | A × B | A set whose members are all possible ordered pairs (a,b) |
Relative complement | - or \ | A - B | All members from A that are not B. Belong to their union, but not intersection. |
Symmetric difference | ∆ or ⊖ | A ∆ B | Members belong to A or B, but not their intersection |
Set Theory Relations...
Relation | Notation | Syntax | English |
---|---|---|---|
Member / element of | ∈ | x ∈ A | x "is a member of" set A |
Not member / element of | ∉ | x ∉ A | x "is not a member of" set A |
Subset | ⊆ | A ⊆ B | set A "is a subset of" set B |
Superset | ⊇ | A ⊇ B | set A "is a superset of" set B |
Proper subset | ⊂ | A ⊂ B | Is a subset, and has less members than B |
Proper superset | ⊃ | A ⊃ B | Is a superset, and has more members than B |
Not subset | ⊄ | A ⊄ B | set A "is not a subset of" set B |
Not superset | ⊅ | A ⊅ B | set A"is not a superset of" set B |
Equality | = | A = B | set A "has the same members as" set B |
Set Theory Sets...
Set | Notation | Example | Extra |
---|---|---|---|
Natural numbers | N or ℕ_{1} | 1, 2, 3, 4, 5, 6, 7, 8 | The counting numbers |
Whole numbers | W or ℕ_{0} | 0, 1, 2, 3, 4, 5, 6, 7, 8 | Natural numbers, including zero |
Integers | Z or ℤ | -4, -3, -2, -1, 0, 1, 2, 3, 4 | Whole numbers, including negatives |
Rational numbers | Q or ℚ | -4, -1.5, 1, 0, 2, 3.9, 6 | Integers, including fractions |
Real numbers | R or ℝ | -4, -1.5, 1, 0, 2, 3.9, 6, pi | Rational numbers, including irrational numbers |
Universal set | U | Anything | Contains all sets |
Empty / Null set | Ø or 0 | The set of squares with 5 sides | A set with nothing inside of it |
Boolean Logic...
English | Set Theory | Set Theory | |||
---|---|---|---|---|---|
OR | Logical disjunction | A or B | "or" | A union B (exists in one of the sets) | A ∪ B |
AND | Logical conjunction | A and B | "and" | A intersect B (exists in both of the sets) | A ∩ B |
NOR | Logical NOR | Neither A or B | "neither...nor" | Not in A union B (exists in neither sets) | Not ( A ∪ B ) |
Boolean Logic...
English | Set Theory | Set Theory | |||
---|---|---|---|---|---|
XOR | Exclusive disjunction | A or B, not A and B | A union B, not A intersect B | Symmetric difference | |
NAND | Logical NAND | Not A and B | "not both" | Not A intersect B | NOT ( A ∩ B ) |
XNOR | Logical biconditional | A and B, or not A and B | A intersect B, or not A intersection B | A ∩ B OR NOT ( A ∩ B ) |
Propositional Logic: Proposition Notations
Notation | What It Is | Example |
---|---|---|
P | 1st proposition (left side) | P ∧ Q, "P=black hair & Q=6 feet tall" |
Q | 2nd proposition (right side) | P ∨ Q, "P=white skin & Q=blue eyes" |
Predicate Logic: Quantifiers, Logical Connectives
Name | Type | Notation | Informal | |
---|---|---|---|---|
Universal quantifier | Quantifier | ∀ | "given any" or "for all" | (x) is true for everything |
Existential quantifier | Quantifier | ∃ | "there exists" | (x) is the predication of a property or relation to at least one member of the domain |
Conjunction | Logical connective | ∧, &, · | "and" | |
Disjunction | Logical connective | ∨ | "or" | |
Implication | Logical connective | →, ⇒, ⊃ | "implies", "if...then" | |
Biconditional | Logical connective | ↔, ≡ , = | "equals" | |
Negation (not) | Logical connective | ¬, ~ | "not" | |
Contradiction | Logical connective | ⊥, 0, F | ||
Tautology | Logical connective | ⊤, 1, T |