Mathematical Logic

Logic in simple words means to reason. This reasoning can be a legal opinion or even a mathematical confirmation. Well, you can apply certain logic in Mathematics as well and solve mathematical logic problems. Some of the basic mathematical logical operators that you can use in your day to day life are conjunction, disjunction, and negation. You denote these mathematical logic symbols as, ^ for representing conjunction, v for representing disjunction, and for representing negation. In this article, let us discuss some of the basic mathematical logic, mathematical logic formulas along with the truth table and some math logic examples with answers.

Classification Of Mathematical Logic

The mathematical logic can be subdivided into four different fields which are as follows:

  1. Set Theory

  2. Recursion Theory

  3. Model Theory

  4. Proof Theory

Mathematical Logical Operators

There are three basic mathematical logical operators that you use in mathematics. These are:

  1. Conjunction or (AND)

  2. Disjunction or (OR)

  3. Negation or (NOT)

Now, let us take a look at all these mathematical logical operators in detail.

Mathematical Logic Formulas

Conjunction or (AND)

You can easily join two mathematical logic statements by using the AND operand. It is also called as a conjunction. You can represent it in the symbol form as ∧. In this operator, if either of the statements is false, then the result is false. If both the statements are true, then the result is true. The inputs can be two or more, but the output you get is just one.

Truth Table Of The Conjunction (AND) Operator

Input A

Input B

Output A AND B (A ∧ B)

True 

True 

True

True 

False

False 

False 

True 

False 

False 

False 

False 

Disjunction or (OR)

You can join two statements easily with the help of the OR operand. It is also called as disjunction. You can represent it in the symbolic form as ∨. In this operator, if either of the statements is true, then the result you get is true. If both the statements are false, then the result is false. It consists of two or more inputs but only one output.

Truth Table Of The Disjunction (OR) Operator

Input A

Input B

Output A OR B (A V B)

True 

True 

True

True 

False

True

False 

True 

True

False 

False 

False 

Negation or (NOT) 

Negation is an operator that gives the opposite statement of the statement which is given. It is also called as NOT and is denoted by ∼. It is an operation which would give the opposite result. When the input is true, the output you get is false. When the input is false, the output you get is true. It consists of one input and one output.

Truth Table Of The Negation (NOT)

Input 

Output

A

Negation A (∼ A)

True 

False

False 

True

Mathematical Logics Problems

Now that you know about the mathematical logic formulas, let us take a look at math logic examples with answers.

Example 1:

Construct a truth table for the values of conjunction for the following given statements:

r: x is an odd number

s: x is a prime number

Solution:

Given:

r: x is an odd number

s: x is a prime number

Since each statement given represents an open sentence, the truth value of r∧s would depend on the value of the variable x. Since there are an infinite number of replacement values for x, you cannot list all the truth values for r∧s in the truth table. However, you can find the truth value of r∧s for the given values of x as follows:

If x = 3, r is true, and s is true. Hence, the conjunction r∧s is true.

If x = 9, r is true, and s is false. Hence, the conjunction r∧s is false.

If x = 2, r is false, and s is true. Hence, the conjunction r∧s is false.

If x = 6, r is false, and s is false. Hence, the conjunction r∧s is false.

x Value

Input r

Input s

Output r AND s (r∧s)

3

True 

True 

True

9

True

False 

False 

2

False 

True 

False 

6

False 

False 

False 


Example 2:

Find the negation of the given statement: 

Number 4 is an even number

Solution:

Consider P to be the given statement

Hence, P = 4 is an even number.

Therefore, the negation of the statement is given as

∼S = 4 is not an even number.

Hence, the negation of the statement is that 4 is not an even number.

FAQ (Frequently Asked Questions)

1. What is Mathematical Logic?

Mathematical logic is primarily about providing a framework to communicate and explain results to each other. With the help of some commonly accepted definitions and understanding rigorously what it means when something is true, false, assumed, etc., you can explain and prove the reasons behind the things being the way they are.

2. What is the Importance of Mathematical Logic in Maths?

Mathematical reasoning depends on logic and the rules of inference in logic for drawing inferences, make deductions, and form valid proofs for conjectures becoming theorems. Logic is, therefore, of fundamental importance in maths.