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Methods of Proof: Vacuous Proof

THIS IS __NOT__ HOMEWORK! This is a practice problem from our textbook. Class link: Discrete Math I

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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 $\displaystyle R$ is.

For #2 are that two pair in $\displaystyle R$ that look like $\displaystyle (a,b),~(b,c)$.

The answer is no. So $\displaystyle (a,b)\in R\text{ and }(b,c)\in R$ is a false statement.

For #3. The statement "It false then anything" is always true.

So if $\displaystyle (a,b)\in R\text{ and }(b,c)\in R$ then $\displaystyle (a,c)\in R$ is TRUE.

Because **A false statement implies any statement.**

Re: Methods of Proof: Vacuous Proof