MATH135 Lecture Notes - Lecture 2: Open Formula, Rational Number, Prime Number
14 May 2018
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MATH 135 Fall 2015: Extra Practice Set 2
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. Suppose Sand Tare two sets. Prove that if S∩T=S, then S⊆T. Is the converse true?
2. Give an example of three sets A,B, and Csuch that B6=Cand B−A=C−A.
3. Prove the following two quantified statements.
(a) ∀n∈N,n+ 1 ≥2
(b) ∃n∈Z,(5n−6)
3∈Z
Recommended Problems
1. Let Sand Tbe any two sets in universe U. Prove that (S∪T)−(S∩T)=(S−T)∪(T−S).
2. Let A={n∈Z: 2 |n}and B={n∈Z: 4 |n}. Prove that n∈(A−B) if and only if n= 2kfor
some odd integer k.
3. Let A={1,{1,{1}}}. List all the elements of A×A.
4. Let a, b, c ∈Z. Is the following statement true? Prove that your answer is correct.
a|bif and only if ac |bc.
5. For each of the following statements, identify the four parts of the quantified statement (quantifier,
variable, domain, and open sentence). Next, express the statement in symbolic form and then write
down the negation of the statement (when possible, without using any negative words such as “not”
or the ¬symbol, but negative math symbols like 6=,-are okay).
(a) For all real numbers xand y,x6=yimplies that x2+y2>0.
(b) For every even integer aand odd integer b, a rational number ccan always be found such that
either a<c<bor b<c<a.
(c) There is some x∈Nsuch that for all y∈N,y|x.
(d) There exist sets of integers X, Y such that for all sets of integers Z,X⊆Z⊆Y.
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Document Summary
Math 135 fall 2015: extra practice set 2. 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: suppose s and t are two sets. Prove that if s t = s, then s t .
