## CSCI 2824 Lecture # 6 NotesIn the previous lectures, we studied implications. This lecture will focus on implications and start chapter 2 on proof techniques. Implications: Converse and Contrapositive. Negation of an implication. Primer on writing proofs (Section 2.1).
## Implications: Contrapositive and ConverseIn simple terms, if is an implication then is its *contra-positive*.is its *converse*.
This is a very common source of confusion. Examples: For all integers , if are both odd then is even. Contrapositive: For all integers , If is not even then both cannot be odd. (Is this true?) Converse: For all integers , If is even then are both odd. (Is this true?)
If is odd then is odd. Contrapositive: If is not odd then is not odd. (Is this true?) Converse: If is odd then is odd. (Is this true?)
Proof: We are asked to prove in effect that is logically equivalent to . We write down the truth table to verify this fact. QED. ## The Converse TrapProving the converse of a statement is often a trap that even experienced researchers can fall into. Note that proving the converse of an implication in no way convinces us of the truth of the implication. Let us take examples from real life to see how silly it could be: **Statement**If a person is underage and consumes of alcoholic beverages then he/she is breaking the law.**Converse**If a person is breaking the law then he/she consumes alcoholic beverages.
Does the converse ring true? ## Primer on Proofs and Mathematical WritingThis lecture, we will warm up by practicing some proofs and the right way to express those proofs. A proof is meant to convince people that some mathematical fact is true. It is meant to be read and judged by ones peers. Therefore writing proofs out well is an important skill for discrete mathematics. ## Example # 0Let us prove the following statement.
Dear reader letter
Dear reader, You have expressed criticism to my theorem in the annals of CSCI 2824 that can be written as If you were kind enough to provide me with any number that we will call , then there are two cases: You provided me with an odd number . The antecedent of the implication is false. Therefore, the implication itself holds true. I have nothing else to say to you in this case. If, however, you were clever enough to provide me with an even value of , we may of course write it in the form for some number . Also, by elementary algebra. As you know divides which is the same as . Therefore is even.
Thus, in no case, can you produce a number that satisfies the antecedent but violates the consequent (). I remain your faithful prover of claims. A fond Note to class: Write the proof in more sensible language. Proof
QED. ## Example# 1
Proof Attempt 1
Let us attempt one more proof of this: Proof Attempt 2
Since is prime, and , must necessarily be odd. Since is odd, it must be the case that is even. If is even, we have proved previously that is also even. Therefore, is even. QED??
Are there any flaws in either of these proofs? Do they convince you of the truth of our “claim”? ## Example #2
Write this down in logical notation? Let us look at a proof: Proof
Since is odd, it can be written as for some . Since is odd, it can be written as too. Therefore . But is an even number. Therefore, is even. QED.
Is there anything wrong with the proof above? Now let us look at a related claim:
Exercise: Write this statement down in Predicate Logic. Is this a true statement? Proof
Since is odd, it can be written as for some . Since is odd, it can be written as too. Therefore . . Therefore, . QED.
Can you correct the demonstrations above? What went wrong? ## More Complicated Proof
Before we dive further, what is a composite number? Definition: Composite Number
A natural number is composite if it can be written as with . The provision that is very important. Or else any number can trivially be written as , with and . Now, back to our claim.
Proof
We can write as a product of . Therefore is a composite number. QED??
Have we really proved the claim above? |