Logic

Constants

Symbol	LaTeX	Notation
True	\top	\top
’’	\mathsf{T}	\mathsf{T}
’’	\operator{True}	\operatorname{True}
False	\bot	\bot
’’	\mathsf{F}	\mathsf{F}
’’	\operatorname{False}	\operatorname{False}

Logical Operators

Symbol	LaTeX	Notation
And	p \land q	p \land q	Conjunction
’’	p \operatorname{and} q	p \operatorname{and} q
Or	p \lor q	p \lor q	Disjunction
’’	p \operatorname{or} q	p \operatorname{or} q
Xor	p \veebar q	p \veebar q	Exclusive Or
Nand	p \barwedge q	p \barwedge q	Not And
Nor	p \char"22BD q	p \char"22BD q	Not Or
Not	\lnot p	\lnot p	Negation
’’	\operatorname{not} p	\operatorname{not} p
Equivalent	p \iff q	p \iff q
’’	p \Leftrightarrow q	p \Leftrightarrow q
Implies	p \implies q	p \implies q
’’	p \Rightarrow q	p \Rightarrow q
Proves	p \vdash q	p \vdash q
Entails	p \vDash q	p \vDash q
Satisfies	p \models q	p \models q

Quantifiers

["ForAll", condition, predicate]

The ForAll function represents the universal quantifier.

The condition is the variable or variables that are being quantified over or the set of elements that the variable can take.

The predicate is the statement that is being quantified.

The condition and the predicate are separated by a comma, a colon, or a vertical bar. The predicate can also be enclosed in parentheses after the condition.

\forall x, x + 1 > x

$$\forall x, x + 1 > x$$

\forall x: x + 1 > x

$$\forall x: x + 1 > x$$

\forall x\mid x + 1 > x

$$\forall x\mid x + 1 > x$$

\forall x( x + 1 > x)

$$\forall x( x + 1 > x)$$

\forall x \in \R, x + 1 > x

$$\forall x \in \R, x + 1 > x$$

["ForAll", "x", ["Greater", ["Add", "x", 1], "x"]]

["ForAll", ["Element", "x", "RealNumbers"], ["Greater", ["Square", "x"], 0]]

["Exists", condition, predicate]

The Exists function represents the existential quantifier.

The condition is the variable or variables that are being quantified over, and the predicate is the statement that is being quantified.

The condition and the predicate are separated by a comma, a colon, or a vertical bar. The predicate can also be enclosed in parentheses after the condition.

\exists x, x^2 = 1

$$\exists x, x^2 = 1$$

\exists x: x^2 = 1

$$\exists x: x^2 = 1$$

\exists x\mid x^2 = 1

$$\exists x\mid x^2 = 1$$

\exists x( x^2 = 1)

$$\exists x( x^2 = 1)$$

["Exists", "x", ["Equal", ["Square", "x"], 1]]

["Exists", ["Element", "x", "RealNumbers"], ["Equal", ["Square", "x"], 1]]

["ExistsUnique", condition, predicate]

The ExistsUnique function represents the unique existential quantifier.

\exists! x, x^2 = 1

$$\exists! x, x^2 = 1$$