A Quick Look at Quantified Statements

1. A Quick Look at Quantified Statements

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

3. 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?

4. 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.

5. Mathematical Convention for Interpreting Statements “for all… there exists…” (AE) {a0, a1, a2, a3, a4, …} {b0, b1, b2, b3, b4, …}

6. Mathematical Convention for Interpreting Statements “there exists… for all…” (EA) {a0, a1, a2, a3, a4, …} {b0, b1, b2, b3, b4, …}

7. 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