Math Logic and Statements

Statements and Truth Values

• A statement is a sentence that is either true or false, but not both.
• For example, "For all real numbers $a$, $a^2 >= 0$" is a statement because it is always true.
• On the other hand, "$x + 10$" is not a statement because it is an expression that can be true or false depending on the value of $x$.

Truth Values

• The truth value of a statement is either 'T' (true) or 'F' (false).
• For example, the statement "$3$ is a prime number" has a truth value of 'T' because it is true.
• The statement "$1$ is a prime number" has a truth value of 'F' because it is false.

Conditional Statements

• A conditional statement is a statement that can be written in the form "if $p$ then $q$".
• For example, "If $2 = 2$ then $2$ is prime" is a conditional statement.
• The symbol '->' is often used to represent "if and then" in conditional statements.

Conjunctions

• A conjunction is a statement that combines two or more statements using the word "and".
• For example, "$x$ is odd and $15$ is composite" is a conjunction.
• The truth value of a conjunction is 'T' only if all the individual statements are true.

Disjunctions

• A disjunction is a statement that combines two or more statements using the word "or".
• For example, "$3 = 0$ or $2$ is even" is a disjunction.
• The truth value of a disjunction is 'T' if at least one of the individual statements is true.