Thinking Mathematically Statements Negations and Quantified Statements Statements

Thinking Mathematically Statements, Negations, and Quantified Statements

Statements A “statement” is a sentence that is either “true” or “false” but not both at the same time. Examples “Miami is a city in Florida” is a true statement. “Two plus two equals five” is a false statement. “Today is Friday” is a statement that is sometimes true and sometimes false, but never both true and false at the same time. “Go to the grocery store” is not a statement. It is a command is neither true nor false.

Statements - “Negation” The “negation” of a statement is another statement that has the opposite “truth value. ” That is when a statement is true its negation is false and when the statement is false its negation is true. Examples “Miami is not a city in Florida” is the negation of the statement “Miami is a city in Florida”. “Two plus two is not equal to five” is the negation of the statement “Two plus two equals five”. Be careful! “Two plus two equals four” is a true statement but it is not the negation of “Two plus two equals five”.

Statements - “Symbolism” Just as x can be used as a name for a number, a symbol such as p can be used as a name for a statement. When p is used as a name for a statement the symbols ~p are used as a name for the negation of p. Examples Let p stand for “Miami is a city in Florida. ” Then ~p is the statement “Miami is not a city in Florida. ”

“Quantified” Statements A “quantified” statement is one that says something about “all”, “some”, or “none” of the objects in a collection. Examples “All students in the college are taking history. ” “Some students are taking mathematics. ” “No students are taking both mathematics and history. ”

“Equivalent” Statements In any language there are many ways to say the same thing. The different linguistic constructions of a statement are considered equivalent. Example “All students in the college are taking history. ” “Every student in the college is taking history. ” Example “Some students are taking mathematics. ” “At least one student is taking mathematics. ”

Negating Quantified Statements The negation of a statement about “all” objects is “not all”. “Not all” can often be expressed by “some are not. ” Examples p : All students in the college are taking history. ~p : Some students in the college are not taking history.

Negating Quantified Statements The negation of a statement about “some” objects is “not some”. “Not some” can often be expressed by “none” or “not any. ” Examples p : Some students are taking mathematics. ~p : None of the students are taking mathematics.