Statements and Logical Operations A statement is a mathematical expression that is either true or false. Statements can be combined using logical operato...
A statement is a mathematical expression that is either true or false. Statements can be combined using logical operators to form complex statements.
Examples:
5 + 3 = 8 (True)
2 + 4 = 6 (True)
7 - 1 = 6 (False)
Logical operators are used to combine statements in different ways. Some of the most common operators are:
AND (∧): An AND statement requires both statements to be true for the overall statement to be true.
OR (∨): An OR statement requires at least one of the statements to be true for the overall statement to be true.
NOT (¬): A NOT statement negates a statement, meaning it is true if and only if the original statement is false.
Combining statements with logical operators:
Conjunction (∧): Combine two statements using an AND operator. For example, the statement "5 + 3 = 8" is a conjunction of the two statements "5 + 3 = 7" and "7 = 8".
Disjunction (∨): Combine two statements using an OR operator. For example, the statement "2 + 4 = 6" is a disjunction of the two statements "2 + 4 = 5" and "5 = 6".
Negation (¬): Combine a statement with a NOT operator. For example, the statement "7 - 1 = 6" is a negation of the statement "7 - 1 = 5".
Rules for combining statements:
The order of statements does not matter in a conjunction. For example, the statements "5 + 3 = 8" and "3 = 8" are equivalent.
The order of statements matters in an OR statement. For example, the statement "2 + 4 = 6" is equivalent to the statement "5 = 6".
By understanding statements and logical operators, you can combine them to solve mathematical problems and prove mathematical theorems