Proposition [ English ]

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In Discrete Mathematics, a proposition is a declarative statement that is either true or false, but not both at the same time.

Definition

A proposition is a sentence that has a definite truth value:

• True (T) or
• False (F)

There is no ambiguity in a proposition.

Examples of Propositions

• “2 + 3 = 5” → True
• “7 is an even number” → False
• “The sun rises in the east” → True
• “India is in Europe” → False

Each of the above statements can be clearly classified as true or false.

Non-Examples (Not Propositions)

The following are not propositions because they do not have a definite truth value:

• “What is your name?” (question)
• “Close the door.” (command)
• “x + 5 = 10” (depends on the value of x)
• “This statement is false.” (paradox)

Types of Propositions

1. Simple (Atomic) PropositionA single statement with no logical connectives.Example:

   • “5 is a prime number.”

2. Compound PropositionFormed by combining simple propositions using logical operators such as AND (∧), OR (∨), NOT (¬), IF–THEN (→).Example:

   • “5 is a prime number and 10 is even.”