MATH135 Lecture Notes - Lecture 3: Integer Factorization, Contraposition, Rational Number
14 May 2018
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MATH 135 Fall 2015: Extra Practice Set 3
These problems are for extra practice and are not to be handed. Solutions will not be posted but, unlike
assignment problems, they may discussed in depth on Piazza.
•The warm-up exercises are intended to be fairly quick and easy to solve. If you are unsure about any
of them, then you should review your notes and possibly speak to an instructor before beginning the
corresponding assignment.
•The recommended problems supplement the practice gained by doing the corresponding assignment.
Some should be done as the material is learned and the rest can be left for exam preparation.
•A few more challenging extra problems are also included for students wishing to push themselves
even harder. Do not worry if you cannot solve these more difficult problems.
Warm-up Exercises
1. Let xbe a real number. Prove that if x3−5x2+ 3x6= 15 then x6= 5.
2. In the proof by contradiction of Prime Factorization (PF), why is it okay to write r≤s?
3. What is the remainder when −98 is divided by 7?
Recommended Problems
1. Let xand ybe integers. Prove that if xy = 0 then x= 0 or y= 0.
2. Let aand bbe integers. Prove that (a|b∧b|a)⇐⇒ a=±b.
3. Prove that an integer is even if and only if its square is an even integer.
4. Let xand ybe integers. Prove or disprove each of the following statements.
(a) If 2 -xy then 2 -xand 2 -y.
(b) If 2 -yand 2 -xthen 2 -xy.
(c) If 10 -xy then 10 -xand 10 -y.
(d) If 10 -xand 10 -ythen 10 -xy.
5. Consider the following statement.
For all x∈R, if x6+ 3x4−3x < 0, then 0 <x<1.
(a) Rewrite the given statement in symbolic form.
(b) State the hypothesis of the given statement.
(c) State the conclusion of the given statement.
(d) State the converse of the given statement.
(e) State the contrapositive of the given statement.
(f) State the negation of the given statement without using the word “not” or the ¬symbol (but
symbols such as 6=, -, etc. are fine).
(g) Prove or disprove the given statement.
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Document Summary
Math 135 fall 2015: extra practice set 3. These problems are for extra practice and are not to be handed. Solutions will not be posted but, unlike assignment problems, they may discussed in depth on piazza: the warm-up exercises are intended to be fairly quick and easy to solve. If you are unsure about any of them, then you should review your notes and possibly speak to an instructor before beginning the corresponding assignment: the recommended problems supplement the practice gained by doing the corresponding assignment. Some should be done as the material is learned and the rest can be left for exam preparation: a few more challenging extra problems are also included for students wishing to push themselves even harder. Do not worry if you cannot solve these more di cult problems. Warm-up exercises: let x be a real number. Recommended problems: let x and y be integers.
