Propositions and Logical Operations

• Logic is being studied

Definitions and Propositions

• A proposition (or statement) is a sentence that is either true or false, but not both.
• A predicate (or propositional function) is a sentence that contains one or more variables that become a proposition when specific values are substituted for these variables.

Logical Operations

• $p$ and $q$ are two propositions.
• The conjunction of $p$ and $q$, denoted $p \land q$ (read "$p$ and $q$"), is the proposition that is true if $p$ and $q$ are both true, and false otherwise.
• The disjunction of $p$ and $q$, denoted $p \lor q$ (read "$p$ or $q$"), is the proposition that is true if at least one of the two propositions is true, and false otherwise.
• The negation of $p$, denoted $\neg p$ (read "not $p$"), is the proposition that is true if $p$ is false, and false if $p$ is true.
• The exclusive or of $p$ and $q$, denoted $p \oplus q$ (read "$p$ exclusive or $q$"), is the proposition that is true if exactly one of the two propositions is true, and false otherwise.
• The implication of $p$ and $q$, denoted $p \rightarrow q$ (read "if $p$, then $q$" or "$p$ implies $q$"), is the proposition that is false only if $p$ is true and $q$ is false, and true in all other cases.
• In the implication $p \rightarrow q$, $p$ is the hypothesis, the antecedent or the premise, and $q$ is the conclusion or the consequent.
• The reciprocal of $p \rightarrow q$ is $q \rightarrow p$.
• The contrapositive of $p \rightarrow q$ is $\neg q \rightarrow \neg p$.
• The inverse of $p \rightarrow q$ is $\neg p \rightarrow \neg q$.
• The biconditional of $p$ and $q$, denoted $p \leftrightarrow q$ (read "$p$ if and only if $q$"), is the proposition that is true if $p$ and $q$ have the same truth value, and false otherwise.

Priority of Logical Operations

• In the absence of parentheses, logical operations are evaluated in the following order of priority:
• Negation ($\neg$)
• Conjunction ($\land$)
• Disjunction ($\lor$)
• Implication ($\rightarrow$)
• Biconditional ($\leftrightarrow$)