When you show the statement is true/false, you pick a number for the part of the proof. (Ex. x, y, z). In the main part of the proof, a number I picked should be expressed as a different number from the one in the original statement. If the number's symbol was x, then I should use the number, for example, x' to indicate that this is different number. Then, when I conclude, I can use the original symbol to generalize. The below is the given statement in quiz #4.
So, basically, to prove the above statement, in the process of proving, I should have used other related symbol instead of z in the original statement. If it wasn't this quiz, I probably wouldn't realize and keep doing in a wrong way in my proof. Now, the following is the proof that I rewrote with 'the correct' symbols.
Assume x and y are real numbers. # x, y are typical generic real numbers.
Assume x > y. # antecedent.
Let z is a real number. # def'n of z'. z' is a real number st x > z and x > y.
Then, x > z'.
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# Prove there is z' satisfies x > z'.
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Then, z' > y.
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# Prove there is z' satisfies z' > y.
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Then, x > z and z' > y # by previous proofs.
Then,
Then,
Then,
Sorry for the unmatched line in the last three steps of my proofs! I tried to match the line, but since I copied the equation as a picture, the size won't change. I tried to type the whole proof using LaTex but it wasn't pretty in the end. Anyway, the bottom line is that now I remember for sure to use different symbols in my main part of proof!
Remember that, when we're dealing with ∃z∈ℝ, instead of just saying 'Let z be a real number', we need to give a specific example:
ReplyDeleteLet z' = #whatever-we-need-it-to-equal
#Prove that z'∈ℝ
Then, z'∈ℝ #now that we've proven it
#Now we prove that x>z'
Then, x>z' #finally we can actually state this
All the while, z' means that we're referring to the 'whatever-we-need-it-to-equal', not some random—generic—real number z.
Keep up the good work!