Introduction to Mathematical Logic
Last Updated :
06 Nov, 2024
Mathematical logic deals with the logic in mathematics. Mathematical logic operators and laws define various statements in their mathematical form. In this article, we will explore mathematical logic along with the mathematical logic operators and types of mathematical logic. We will also solve some examples related to mathematical logic.
What is Mathematical Logic?
The study of mathematical logic in mathematics is called mathematical logic. The basic mathematical logic used are the conjunction (∧), disjunction (∨), and negation (¬). Some other mathematical logics are implication and double implication.
Mathematical Logic
Mathematical Logic Operators
The basic mathematical logic operators are:
- Conjunction
- Disjunction
- Negation
- Implication
- Double Implication
Conjunction
In mathematical logic conjunction of two statements results in true when both the statements are true otherwise false. Conjunction is also known as AND operator and is represented by ∧.
Disjunction
In mathematical logic disjunction of two statements results in false if both the statements are false otherwise true. Disjunction is also known as OR operator and is represented by ∨.
Negation
In mathematical logic negation of two statements results in the not of the given statement i.e., if the statement is true it results in false and if the statement is false it results in true. Negation is also known as NOT operator and is represented by ~ or ¬.
Implication
In mathematical logic implication of two statements results in false if the first statement is true and second statement is false otherwise true. Implication is also known as conditional operator and is represented by → or ⇒. Implication X→Y is read as If X and then Y.
Double Implication
In mathematical logic double implication of two statements results in true when either both statements are true or both statements are false. Double implication is also known as biconditional operator and is represented by ↔ or ⇔. Double implication X↔ Y is read as Y iff X or Y if and only if X.
Some of the basic mathematical formulas are listed below:
Formula Names | Mathematical Logic Formula |
|---|
Identity Law | |
Domination Law | |
Idempotent law | |
Double Negation Law | |
Commutative Law | - (a ∧ b) ≣ (b ∧ a)
- (a ∨ b) ≣ (b ∨ a)
|
Associative Law | - (a ∧ b) ∧ c ≣ a ∧ (b ∧ c)
- (a ∨ b) ∨ c ≣ a ∨ (b ∨ c)
|
Distributive Law | - a ∧ (b ∨ c) ≣ (a ∧ b) ∨ (a ∧ c)
- a ∨ (b ∧ c) ≣ (a ∨ b) ∧ (a ∨ c)
|
De Morgan Law | - ¬ (a ∧ b) ≣ ¬a ∨ ¬b
- ¬ (a ∨ b) ≣ ¬a ∧ ¬b
|
Absorption Law | - a ∧ (a ∨ b) ≣ a
- a ∨ (a ∧ b) ≣ a
|
Negation Law | |
Types of Mathematical Logic
The different types of mathematical logic include:
- Set Theory
- Model Theory
- Proof Theory
- Recursion Theory
Set Theory: Set theory is a part of mathematical logic that deals with the sets which means collection of elements. The set theory is the theory consisting of sets, sets formulas and many more.
Model Theory: Model theory is a part of mathematical logic that deals with the models of different theories of mathematics. The model theory provides different models describing the complex theories making it easy to understand.
Proof Theory: Proof theory is a part of mathematical logic that deals with the proofs. The mathematical proofs provide easy analysis of different mathematical methods.
Recursion Theory: Recursion theory is a part of mathematical logic used to construct computable functions, Turing machines and recursively enumerable sets.
Mathematical Logic Truth Table
The truth table in mathematical logic is a table which takes inputs and provides output when a logic is applied to it. The truth table for different mathematical logic operators are given below.
Negation
The truth table for negation is given below.
Conjunction
The truth table for conjunction is given below.
A | B | A∧B |
|---|
True | True | True |
True | False | False |
False | True | False |
False | False | False |
Disjunction
The truth table for disjunction is given below.
A | B | A∨B |
|---|
True | True | True |
True | False | True |
False | True | True |
False | False | False |
Implication
The truth table for implication is given below.
A | B | A→B |
|---|
True | True | True |
True | False | False |
False | True | True |
False | False | True |
Double Implication
The truth table for double implication is given below.
A | B | A↔B |
|---|
True | True | True |
True | False | False |
False | True | False |
False | False | True |
Mathematical Logic Solved Examples
Example 1: Consider the statement x < 5 → x - 2 < 5 is true or false?
Solution:
If x < 5 is true then, x - 2 < 5 is also true.
T → T is true
So, the given statement x < 5 → x - 2 < 5 is true.
Example 2: For given two statements compute the truth table for conjunction.
P: a is divisible by 4
Q: a is divisible by 10
Solution:
Given the two statements P and Q.
P: a is divisible by 4
Q: a is divisible by 10
Value of a | P | Q | P ∧ Q |
|---|
20 | T | T | T |
8 | T | F | F |
30 | F | T | F |
5 | F | F | F |
Example 3: Find the negation of the given statement P: It is a rainy day.
Solution:
Given statement,
P: It is a rainy day.
Negation of P = ¬ P: It is not a rainy day.
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