What is a proposition?
In logical operations, a proposition is a declarative statement that can either be true or false that helps us make a defined decision on a statement. It is a fundamental concept in logical reasoning and decision making which finds applications in various fields, including mathematics, philosophy, and computer science.
Truth values of a proposition
A truth value is a representation in logical operations to determine the correctness of a statement. It has two types:
True: Our statement is true and refers to a correct claim. It is represented by T.
False: Our statement is false and refers to an incorrect claim. It is represented by F.
Examples
Let's discuss some examples of propositions to clear our concept further.
The sun rises in the east.
This proposition has a true truth value as it is a correct statement.
5 > 15
This proposition has a false truth value as it is a mathematically incorrect expression.
x + 1 =10
This is not a proposition, as the value of x is not defined, so the result of the above statement is undefined instead of having a true or false value.
Types of propositions
There are two types of propositions based on their divisibility.
1. Atomic proposition
An atomic proposition is a type of proposition that exists in the simplest form and cannot be divided into further statements.
Example
The statement "It is raining" is an atomic proposition with either a true or false truth value. This statement cannot be further divided into sub-statements and does not depend on the truth or falsity of any other proposition.
2. Compound proposition
A compound proposition is a type of proposition that consists of a combination of multiple atomic propositions. These atomic propositions our combined using connectives such as AND, XOR, and OR. We will be studying about these connectives in the later part of this Answer.
Example
The statement "It is raining, and the floor is wet" is a compound proposition that consists of two atomic propositions:
It is raining
The floor is wet
These propositions are joined together using the AND connective.
Connectives
A connective is a logical operation that helps establish a relationship between two propositions. To understand these better, suppose we have two propositions: m and n. Let's study about most common connectives applied to these propositions now.
Connective | Representation | Explanation |
AND | It is represented by "∧" between the propositions: m ∧ n. | It returns true if all input truth values are true else returns false for all cases. |
OR | It is represented by "∨" between the propositions : m ∨ n. | It returns true if any one of the input truth values is true else returns false. |
Negation | It is represented by "~" with the proposition: ~m. | It returns the truth value opposite to the input truth value. |
Implication | It is represented by "→" between the propositions : m → n. | It returns false when the first input truth value is true and the second is false. For all other cases it returns true. |
Bi-conditional | It is represented by "↔" between the propositions : m ↔ n. | It returns true if the input truth values are same else returns false for all other cases. |
Truth table
A truth table is a representation of a proposition to show all possible truth values. It helps us understand the formation of compound propositions after connectives are used with two or more atomic propositions.
Example
Let's suppose we have 2 propositions: m and n. Let's represent their AND operation using a truth table.
Representation of AND operation
m | n | m ∧ n |
T | T | T |
T | F | F |
F | T | F |
F | F | F |
Conclusion
Propositions are the building blocks of logical reasoning that help us express statements with clear truth values. They find their applications in various fields to establish theorems, solve proofs, make decisions, and analyze statements' correctness. By understanding the representation and types of propositions, we can easily represent our statements with a defined decision.
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