Logic

Legends

Propositional Symbols

Symbol	Meaning
¬P	NOT P (negation)
P ∧ Q	P AND Q (conjunction)
P ∨ Q	P OR Q (inclusive OR)
P ⊕ Q	Exclusive OR (exactly one true)
P → Q	If P then Q (implication)
P ↔ Q	P iff Q (biconditional)
⊤ / ⊥	Always true / always false

Predicate Symbols

Symbol	Meaning
∀x	For all x (universal quantifier)
∃x	There exists an x (existential quantifier)
P(x)	Predicate P applied to argument x
R(x,y)	Relation R between x and y
x = y	Equality: x and y refer to the same object
f(x)	Function term f applied to x
¬P(x)	Negation of predicate statement P(x)
∀x P(x)	P holds for every x in the domain
∃x P(x)	P holds for at least one x in the domain

Quantifier Symbols

Symbol	Meaning
∀	Universal quantifier: “for all”
∃	Existential quantifier: “there exists”
∀x P(x)	P(x) is true for every x in the domain
∃x P(x)	There is at least one x in the domain where P(x) is true
∀x∈D P(x)	For every x in set/domain D, P(x) holds
∃x∈D P(x)	There exists an x in set/domain D such that P(x) holds
¬∀x P(x)	Not all x satisfy P(x) (equivalent to ∃x ¬P(x))
¬∃x P(x)	No x satisfies P(x) (equivalent to ∀x ¬P(x))

Set Symbols

Symbol	Meaning
∈	Is an element of
∉	Is not an element of
⊆	Subset of (possibly equal)
⊂	Proper subset of
⊇	Superset of (possibly equal)
∪	Union of sets
∩	Intersection of sets
A \ B	Set difference: elements in A not in B
A △ B	Symmetric difference
∅	Empty set
\|A\|	Cardinality (number of elements in A)
𝒫(A)	Power set of A (all subsets of A)

Relation Symbols

Symbol	Meaning
xRy	x is related to y by relation R
(x,y) ∈ R	Ordered pair (x,y) belongs to relation R
R ⊆ A×A	R is a relation on set A
R⁻¹	Inverse relation of R
R∘S	Composition of relations R and S
Idₐ	Identity relation on A
Reflexive	∀x, xRx
Symmetric	∀x∀y, xRy → yRx
Antisymmetric	∀x∀y, (xRy ∧ yRx) → x=y
Transitive	∀x∀y∀z, (xRy ∧ yRz) → xRz

Function Symbols

Symbol	Meaning
f: A → B	Function f maps elements from domain A to codomain B
f(x)	Value of function f at input x
x ↦ f(x)	Rule notation: x maps to f(x)
dom(f)	Domain of f (all allowed inputs)
codom(f)	Codomain of f (declared output set)
im(f)	Image/range of f (actual outputs produced)
f⁻¹	Inverse function (exists when f is bijective)
(f∘g)(x)	Composition: apply g first, then f
Injective	One-to-one: different inputs give different outputs
Surjective	Onto: every codomain element is hit
Bijective	Both injective and surjective

Lab

Designed and Developed by Franz Ayestaran / Enhanced Pair Programming with OpenAI GPT

Logic Wiki

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Interactive Logic Gates Tutorial

Explore AND, OR, NOT, NAND, NOR, and XOR. Switch inputs to see outputs and compare with each gate truth table.

Gate A B

AND gate from the Gates folder.

Propositional Logic

Predicate Logic

Quantifiers

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