Notes on logic statements and compound statements

Types of sentences

  • Statements: Declarative sentences making a claim, either true or false.

  • Questions: Request information.

  • Commands: Express instructions.

  • Opinions: Express subjective judgments.

  • Exclamations: Convey strong emotion.

What is a statement?

  • A statement is a declarative sentence that is either true or false, but not both. It possesses a truth value (T or F).

  • Examples: "There are twelve months in a year." (True), "Jurassic Park was directed by Steven Spielberg." (Fact).

Open statements and variables

  • An open statement contains a variable, and its truth value depends on the value assigned to that variable.

  • Example: (x - 2 = 6)

    • If x = 8, it's true: (8 - 2 = 6).

    • If x = 9, it's false: (9 - 2 \neq 6).

  • Once a value is assigned, an open statement becomes a truth-valued statement.

Truth values and truth tables

  • The truth value of a simple statement is either true (T) or false (F).

  • Truth tables display all possible truth value combinations for statements and their connectives.

Negation

  • Symbol: \neg P (read 'not P' or 'it is not the case that P').

  • Reverses the truth value of the original statement:

    • If P is true, \neg P is false.

    • If P is false, \neg P is true.

  • Double negation: \neg(\neg P) = P.

Basic symbols for statements and connectives

  • Conjunction: (p \land q) ('p and q')

  • Disjunction: (p \lor q) ('p or q')

  • Conditional: (p \rightarrow q) ('if p, then q' or 'p implies q')

  • Biconditional: (p \leftrightarrow q) ('p if and only if q')

Compound statements vs simple statements

  • Simple statement: Conveys a single idea.

  • Compound statement: Conveys two or more ideas joined by a connective (e.g., and, or, if-then).

Examples of symbolic forms from everyday statements

  • Given:

    • P: 'I will go to the gym today.'

    • Q: 'The temperature outside is very cold.'

  • Examples:

    • (P \land Q) : 'I will go to the gym today and the temperature outside is very cold.'

    • (P \rightarrow Q) : 'If I will go to the gym today, then the temperature outside is very cold.'

Quick recall and practice ideas

  • Identify statements vs. nonstatements.

  • Use truth tables for negation and connectives.

  • Convert sentences to symbolic form.

Real-world relevance and implications

  • Forms the basis of Boolean logic in computer science and digital circuit design.

  • Enhances clear argumentation and analytical thinking.