A Quick Look at Quantified Statements

Why are Quantified Statements Important? The logical structure of quantified statements provides a basis for the construction and validation of proofs.

Consider the quantified statements… “For every positive number b, there exists a positive number a such that a<b. “There exists a positive number a such that for every positive number b, a<b.” Are the statements true or false?

A mathematical interpretation of a quantified statement relies on the quantified structure of the statement. By convention, a quantified statement is interpreted as it is written from left to right.

Mathematical Convention for Interpreting Statements “for all… there exists…” (AE) {b 0, b 1, b 2, b 3, b 4, …} {a 0, a 1, a 2, a 3, a 4, …}

Mathematical Convention for Interpreting Statements “there exists… for all…” (EA) {b 0, b 1, b 2, b 3, b 4, …} {a 0, a 1, a 2, a 3, a 4, …}

Consider the quantified statements… “For every positive number b, there exists a positive number a such that a<b. The statement is true. For any positive number b, we can find a positive number a less than it; for instance, let a=b/2. “There exists a positive number a such that for every positive number b, a<b.” The statement is false. There is no positive number a smaller than every other positive number b; for instance, let b=a/2