Methods of Proof: Vacuous Proof

1 Attachment(s)

THIS IS NOT HOMEWORK! This is a practice problem from our textbook. Class link: Discrete Math I
Attachment 25446

1) "By the definition of R we see that there are no ordered pairsin R where the first entry of one and the second entry of another is the
same..."
I do not understand what the definition of R is, nor what they mean by the first entry of one and the second entry of another is the same.

2) "...so the hypothesis, ..., cannot be true."
Maybe because I do not understand #1 above, I also do not understand this conclusion.

3) "...this shows that the implication is true"
I obviously do not understand this either.

Any clarification would be greatly appreciated. Thank you!

Re: Methods of Proof: Vacuous Proof

Quote:

Originally Posted by sflink

1) "By the definition of R we see that there are no ordered pairsin R where the first entry of one and the second entry of another is the
same..."
I do not understand what the definition of R is, nor what they mean by the first entry of one and the second entry of another is the same.

2) "...so the hypothesis, ..., cannot be true."
Maybe because I do not understand #1 above, I also do not understand this conclusion.

3) "...this shows that the implication is true"
I obviously do not understand this either.

For #1 there is nothing to understand. A relation is a set of ordered pairs. That is exactly what $R$ is.

For #2 are that two pair in $R$ that look like $(a,b),~(b,c)$ .
The answer is no. So $(a,b)\in R\text{ and }(b,c)\in R$ is a false statement.

For #3. The statement "It false then anything" is always true.
So if $(a,b)\in R\text{ and }(b,c)\in R$ then $(a,c)\in R$ is TRUE.
Because A false statement implies any statement.

Re: Methods of Proof: Vacuous Proof

Thank you!