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.