Mathematical logic is the part of mathematics that deals with rules of reasoning and how we can prove things logically. It helps us understand whether a mathematical statement is true or false using clear and step-by-step methods.

There are four main areas in mathematical logic:

• Model theory (studying how statements relate to structures),
• Proof theory (studying how to prove statements),
• Set theory (working with collections of objects), and
• Recursion theory (studying processes that repeat using rules).

Mathematical logic started in the 1800s, combining ideas from philosophy and mathematics. Before that, logic was taught through speaking, writing, and basic reasoning. Over time, it became more mathematical and detailed.

Today, we use mathematical logic to check if arguments and proofs are correct. A proof is a series of steps showing why a statement is true. In a proof, the earlier statements are called premises, and the final statement is the conclusion.

What is Mathematical Logic?

Mathematical logic uses symbols to show different kinds of statements and how they work in reasoning.

• The symbol ‘~’ means not (this is called negation),
• ‘^’ means and (this is called conjunction),
• ‘v’ means or (this is called disjunction).

These symbols help us clearly understand whether a statement is true or false. Using these logical rules, we can tell if a mathematical argument is correct or not.

Mathematical logic isn't just important in math—it’s also widely used in computer science. It helps in designing computer circuits, writing computer programs, and checking if those programs work the way they’re supposed to.

Mathematical Logic Operators

Conjunction: The “AND” operand can be used to connect two statements. It is also identified as a conjunction. It has the symbol “∧“.. If any of the statements in this operator is false, the result will be false. If both statements are true, the outcome will be true. It has a number of inputs but only one output.

A AND B = (A ∧ B)

Truth Table for AND (Conjunction)

Explanation: The result of A AND B is True only when both A and B are True. In all other cases, the result is False.

Disjunction: The “OR” operand can be used to join two statements. It is also referred to as disjunction. It has the symbolic form “∨”. If any of the statements in this operator is true, then the result is true. If both statements are incorrect, the result will be incorrect as well. There are several inputs but only one output.

A OR B = (A ∨ B)

Truth Table for OR (Disjunction)

Explanation: The result of A OR B is True if at least one of the inputs is True. It is False only when both A and B are False.

Negation: Negation is an operator that returns the inverse statement of the given statement. It is also known as NOT, which is denoted by the letter ““∼”. It is an operation that produces the opposite outcome.

Negation A = (∼A)

Truth Table for NOT (Negation)

Explanation: The NOT operation simply reverses the value.

• If A is True, then NOT A is False.
• If A is False, then NOT A is True.

Mathematical Logic Formulas

Logical rules can be used to simplify logical formulas.

Identity laws

\( p\wedge T \equiv p \)

\( p\vee F \equiv p \)

Domination laws

\( p\wedge F \equiv F \)

\( p\vee T \equiv T \)

Idempotent laws

\( p\wedge p\equiv p \)

\( p\vee p\equiv p \)

Double negation law

\( \neg\left ( \neg p\right )\equiv p \)

Commutative laws

\( p\wedge q\equiv q\wedge p \)

\( p\vee q\equiv q\vee p \)

Associative laws

\( \left ( p\wedge q \right )\wedge r\equiv p\left ( q\wedge r \right ) \)

\( \left ( p\vee q \right )\vee r\equiv p\left ( q\vee r \right ) \)

Distributive laws

\( p\wedge \left ( q\vee r \right )\equiv \left ( p\wedge q \right ) \vee \left ( p\wedge r \right ) \)

\( p\vee \left ( q\wedge r \right )\equiv \left ( p\vee q \right )\wedge \left ( p\vee r \right ) \)

De morgan laws

\( \neg\left ( p\wedge q \right )\equiv \neg p \vee \neg q \)

\( \neg\left ( p\vee q \right )\equiv \neg p \wedge \neg q \)

 Absorption laws

\( p\wedge \left ( p\vee q \right )\equiv p \)

\( p\vee \left ( p\wedge q \right )\equiv p \)

 Negation laws

\( p\wedge\neg p\equiv F \)

\( p\vee \neg p\equiv T \)

There are four parts to mathematical logic:

• Model theory
• Proof theory
• Recursion theory
• Set theory

Model theory: A model is a simplified or reduced version of a theory. Models can be thought of as theories with a more narrowly defined scope of explanation. A model is descriptive, whereas a theory is both descriptive and explanatory.

Model theory is the study of the models of various formal theories. A theory is a set of equations with a particular formal logic and signature, whereas a model is a framework that provides a tangibly interpretable interpretation of the theory.

Proof theory: Proof theory is a major branch of mathematical logic that represents proofs as formal mathematical objects, making mathematical techniques easier to analyze.

Proofs are typically presented as inductively defined data structures such as plain lists, boxed lists, or trees that are built according to the logical system’s axioms and rules of inference. As a result, proof theory is syntactic in nature, whereas model theory is semantic in nature. Structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity are some of the major areas of proof theory.

Recursion theory: The computability of functions from positive integers to natural numbers is the subject of classical recursion theory. Using Turing machines, calculus, and other systems, the essential results construct a robust, canonical Class of computable functions with several independent, equivalent characterizations. Two more advanced conclusions are the structure of Turing degrees and the lattice of recursively enumerable sets.

Set theory: Set theory is a branch of mathematical logic that studies sets, which can be described informally as collections of objects. Although any object can be compiled into a set, set theory, set theory symbols as a branch of mathematics, is mostly concerned with those that are relevant to mathematics in general.

Mathematical Logic Truth Table

A truth table is a mathematical table used in logic, specifically in connection with Boolean algebra, Boolean functions, and propositional calculus that lists the functional values of logical expressions on every of their functional arguments, that is, for every value combination taken by their logical variables.

Truth tables, in particular, can be used to determine whether a propositional expression is true for all legitimate input values, i.e., logically valid.

Below mathematical logic, the Truth Table displays the result of combining any two boolean expressions using the AND and OR operators (or the NOT operator).

Mathematical Logic Solved Examples

Example 1: Consider the statement \( x> 0\Rightarrow x+1> 0 \) is this statement true or false?

Solution: To determine its truth value, we must first examine the hypothesis: x> 0

Whatever conclusion we reach, it is a result of the fact that x is positive.

The conclusion is as follows: x+ 1> 0 Because x+1>x>0, this statement must be true.

This implies that the statement is true.

Example 2: Find the truth table for OR (Disjunction) using two statements.

Let’s take two simple statements about a number n:

• A: n is divisible by 2
• B: n is divisible by 5

Now let’s check different values of n and see if these statements are true or false. Then we’ll find the result of A OR B.

Explanation: The result of A OR B is True if at least one of the statements is true.
Only when both are false, the output is False.

Example 3: Find the truth table for OR (Disjunction) using two new statements.

Let’s take these two statements about a number x:

• A: x is greater than 10
• B: x is an even number

Now, we’ll use different values of x and check the truth of each statement, then find the result of A OR B.

Explanation: The output of A OR B is True if at least one of the statements is true.
Only when both are false (like for x = 9), the result is False.

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