Logic symbols

Logic symbols are the symbols used to represent logic in mathematics. There are multiple logic symbols including quantifiers, connectives and other symbols. In this article we will explore all the logic symbols that are useful to represent logical statements in mathematical form. Let’s start our learning on the topic “Logic Symbols.”

Logic symbols

What are Logic Symbols?

The symbols that are used to represent logical statements are called logic symbols. The logic symbols help to convert English statements in the form of mathematical logic. The two main types of mathematical logic are propositional logic and predicate logic. In propositional logic, connective logic symbols are mainly used whereas in predicate logic quantifiers logic symbols are used along with the connectives.

Commonly used logic symbols can either be classified as:

• Quantifiers
• Connectives

Let’s discuss these in detail as follows:

Quantifiers Symbols

Table for some of the most common quantifiers is given below:

Quantifier	Symbol	Meaning	Example
Universal	∀	“For all” or “for every”	∀x (for all x)
Existential	∃	“There exists” or “there is at least one”	∃x (there exists x)
Unique Existential	∃!	“There exists a unique” or “there is exactly one”	∃!x (there exists unique x)
Existential Negative	∄	“There does not exist” or “there is no”	∄x (there does not exist x)
Universal Conditional	∀→	“For every…there is…”	∀x → ∃y (for every x, there is a y)
Existential Conditional	∃→	“There exists…such that…”	∃x → ∀y (there exists x such that for every y)
Existential Unique	∃≡	“There exists exactly one” or “there is a unique”	∃≡x (there exists exactly one x)
Universal Unique	∀≡	“For every…there is exactly one”	∀≡x (for every x, there is exactly one x)

Read more about Predicates and Quantifiers

Connective Symbols

Some examples of connectives are as follows:

Symbol	Name	Meaning	Example
¬	Negation	Negation (NOT)	¬p (not p)
∧	Conjunction	Conjunction (AND)	p ∧ q (p and q)
∨	Disjunction	Disjunction (OR)	p ∨ q (p or q)
→ or ⇒	Implication	Implication (IF…THEN)	p → q (if p, then q)
↔ or ⇔	Equivalence	Equivalence (IF AND ONLY IF)	p ↔ q (p if and only if q)

Truth Table for Connectives

Truth table for all the connectives is given as follows:

p	q	¬p	p ∧ q	p ∨ q	p → q	p ⇔ q
True	True	False	True	True	True	True
True	False	False	False	True	False	False
False	True	True	False	True	True	False
False	False	True	False	False	True	True

Binary Logical Connectives Symbols

Examples of Binary Logical Connectives symbols are as follows:

Symbol Name	Explanation	Example
P ∧ Q	Conjunction (P and Q)	P ∧ Q ≡ Q
P ∨ Q	Disjunction (P or Q)	¬(P ∨ Q) ≡ ¬P ∧ ¬Q
P ↑ Q	Negation of Conjunction (P nand Q)	P ↑ Q ≡ ¬(P ∧ Q)
P ↓ Q	Negative of Disjunction (P nor Q)	P ↓ Q ≡ ¬P ∧ ¬Q
P → Q	Conditional (If P, then Q)	For all P, P → P is a tautology
P ← Q	Converse Conditional (If Q, then P)	Q ← (P ∧ Q)
P ↔ Q	Biconditional (P if and only if Q)	P ↔ Q ≡ (P → Q) ∧ (P←Q)

Other Useful Symbols

Some examples of other useful symbols are as follows:

Symbol	Name	Meaning	Example
∈	Element of	Element of (belongs to)	x ∈ A (x belongs to set A)
∉	Not an element of	Not an element of (does not belong to)	x ∉ A (x does not belong to set A)
⊆	Subset of	Subset of (is a subset of)	A ⊆ B (set A is a subset of set B)
⊇	Superset of	Superset of (is a superset of)	A ⊇ B (set A is a superset of set B)
∅	Empty set	Empty set (null set)	∅ (empty set)
∞	Infinity	Infinity	∞ (infinity)
≡	Identical to	Identical to (equivalence)	a ≡ b (a is equivalent to b)
≈	Approximately equal to	Approximately equal to	a ≈ b (a is approximately equal to b)
≠	Not equal to	Not equal to	a ≠ b (a is not equal to b)
∼	Similar to	Similar to (tilde)	x ∼ y (x is similar to y)
∩	Intersection	Intersection (AND)	A ∩ B (intersection of sets A and B)
∪	Union	Union (OR)	A ∪ B (union of sets A and B)
⊂	Proper subset of	Proper subset of	A ⊂ B (set A is a proper subset of set B)
⊃	Proper superset of	Proper superset of	A ⊃ B (set A is a proper superset of set B)
⊥	Bottom	Bottom (logical falsity or contradiction)	⊥ (logical contradiction)
⊤	Top	Top (logical truth or tautology)	⊤ (logical tautology)
⊨	Entails	Entails (logical consequence)	A ⊨ B (A logically entails B)

Relational Operator Symbols

Some of the relational operators in logic are:

Operator	Symbol	Meaning	Example
Equal to	=	Two values are equal	5 = 5 (true)
Not equal to	≠	Two values are not equal	5 ≠ 3 (true)
Greater than	>	One value is greater than another	5 > 3 (true)
Less than	<	One value is less than another	5 < 3 (false)
Greater than or equal to	≥	One value is greater than or equal to another	5 ≥ 5 (true)
Less than or equal to	≤	One value is less than or equal to another	5 ≤ 3 (false)

Conclusion

In summary, logic symbols are like a special language we use to express ideas very precisely. They help us say things like “for all” or “there exists” and connect different statements together. By using these symbols, we can better understand complex concepts and solve problems in many different areas, like math, science, and philosophy. Learning about logic symbols gives us powerful tools for thinking clearly and solving puzzles in our everyday lives.

Read More,

• Propositional Logic
• Logic Gates
• Difference between Propositional and Predicate Logic

Logic Symbols: FAQs

What are Logic Symbols?

The symbols used to represent logic statements in mathematical logic are called logic symbols.

What are 5 Symbols of Logic?

The 5 symbols of propositional logic are:

• Conjunction
• Disjunction
• Implication
• Equivalence
• Negation

What is ∈ logic symbol?

∈ logic symbol means the element of symbol.

What does P → Q mean?

The statement P → Q means if P then Q i.e., P implies Q.

What is iff Symbol?

The iff symbol or equivalence symbol is ↔ or ⇔.