Discrete Math

Study of mathematical structures that are fundamentally discrete rather than continuous.

Set TheoryLogic and ProofFunctions and Relations
Graph TheoryCombinatoricsNumber Theory
Cryptography and SecurityNetwork AnalysisScheduling and Planning
Breaking Down ProblemsUsing Visuals and Diagrams
Set TheoryLogic and ProofFunctions and Relati...Graph TheoryCombinatoricsNumber TheoryCryptography and Sec...Network AnalysisScheduling and Plann...Breaking Down Proble...Using Visuals and Di...
Basic Concepts

Logic and Proof

Understanding Logic

Logic is the science of reasoning. In discrete math, we use logic to form precise statements and determine if they are true or false.

Statements and Connectives

Statements are sentences that are either true or false, like "The sky is blue." Connectives like "and," "or," and "not" help us combine statements:

  • AND (\(\land\)): Both statements must be true.
  • OR (\(\lor\)): At least one statement is true.
  • NOT (\( eg\)): Reverses the truth value.

Types of Proof

Proofs are logical arguments that show a statement is always true. Common proof techniques include:

  • Direct Proof: Start from known facts and use logic to reach the conclusion.
  • Proof by Contradiction: Assume the opposite of what you want to prove and show a contradiction arises.

Why Logic Matters

Logical thinking and proof skills are essential for programming, solving puzzles, and making sound decisions!

Examples

  • If 'All cats are mammals' and 'Whiskers is a cat,' then 'Whiskers is a mammal.'

  • Using a direct proof to show that the sum of two even numbers is even.

In a Nutshell

Logic helps us build and analyze arguments to determine truth, and proofs show statements are always true.

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Functions and Relations