Last Updated : 05 Mar, 2024
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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
Table of Content
- What are Logic Symbols?
- Quantifiers Symbols
- Connective Symbols
- Other Useful Symbols
- Conclusion
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 andQuantifiers
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 ⇔.
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