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.
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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.
The basic mathematical logic operators are:
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 ∧.
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 ∨.
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 ¬.
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.
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:
Mathematical Logic Formula
Double Negation Law
The different types of mathematical logic include:
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.
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.
The truth table for negation is given below.
