By Wesner

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This, however, is not a proof — a proof is a polished piece of work that shows the connection. It should give your reader no idea as to how easy or hard this was for you to work through and it should not include any wrong paths or other ideas that you had which are not necessary for your argument. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 5. 5 21 Quantifiers Consider the following three statements: • There is a real number x such that x + 1 = 8.

5 ≡ 5 mod 21 5. 5 ≡ 11 mod 6 Before we proceed, we remind you of a definition that was introduced in Chapter 1. That is the definition of a rational number. Since we will use this definition quite a bit, we simply restate it here. A real number x is said to be rational iff ∃a, b ∈ Z, b = 0, x = ab . Now that we have some mathematical terms defined we can begin to work on actually constructing some proofs, but first a few stylistic comments. First off (and we can’t emphasize this enough), proof writing is indeed writing and while it is appropriate to use mathematical symbols in your proofs, as stated before, the writing must be grammatically correct.