CSCI 301 L14 Notes

Lecture 14 - Notes

Proving Non-Conditional Statements

• Know how to prove if-and-only-if statements and equivalence statements
• Know how to prove existentially quantified statements
• Know how to disprove statements in general
• Know how to disprove universally quantified statements and conditional statements by counterexample

Announcements

• A3 due Wednesday.
• Week 3 Survey due Wednedsay night

Proving Non-Conditional Statements

Proving if-and-only-if statements

To prove biconditional statements, we simply use the following evqivalence:

\(P \Leftrightarrow Q \equiv (P \Rightarrow Q) \land (Q \Rightarrow P)\)

Equivalence Statements:

Example: The following are all equivalent:

• \(P:\) \(x\) is even
• \(Q:\) \(2 \mid x\)
• \(R:\) \(x\) is not odd

To prove this, we need to show a (circular) chain of conditionals that verifies that these are either all true or all false. For example, we could prove the equivalences by proving the the following:

• \(P \Rightarrow Q\)
• \(Q \Rightarrow R\)
• \(R \Rightarrow P\)

Or we could show \(P \Leftrightarrow Q\) and \(P \Leftrightarrow R\); or many other combinations are possible, but there must be a complete, closed loop or bidirectional chain.

Disproof

In “real” math, there are some propositions we don’t know the truth value of; important ones are called Conjectures. If something is false, then you can disprove it by proving that its negation is true. The following are all direct consequences of this that follow from our understanding of how to logically negate statements of different forms: