Logic: Definition

The study of logic focuses on the principles of valid reasoning and argumentation. In discrete mathematics, logic is used to analyze and manipulate statements in a formal manner.

Propositional logic is a branch of logic that deals with statements that can be either true or false, but not both.

A proposition is any declarative statement/claim that has a definite truth value (either true or false, but not both.)

For example, the statement "$2 + 2 = 4$" is a true proposition. Conversely, the statement "all prime numbers are odd" is a false proposition (Why is this latter proposition false? Hint: can you name a prime number that isn't odd?). More examples:

• "There are $\infty$-many integers" - true. (Same as claiming: $|\mathbb{Z}| = \infty$ (

  $|\mathbb{Z}| = \infty$

  ).)
• "The number $\sqrt{2}$ is rational" - false. (Same as claiming: $\sqrt{2} \in \mathbb{Q}$ (

  $\sqrt{2} \in \mathbb{Q}$

  ).)
• "The number $\sqrt{4}$ is rational" - true. (Same as claiming: $\sqrt{4} \in \mathbb{Q}$ (

  $\sqrt{4} \in \mathbb{Q}$

  ).)