MATH 135: Extra Practice for Chapter 1
These problems are for extra practice and are not to be handed in. 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 assignments.
Some should be done as the material is learned and the rest can be left for exam preparation.
Warm-up Exercises
1. Determine if the following quantified statements are true or false. No justification is needed.
(a) ∀x ∈ R, sin2 x + cos2 x = 1
(b) ∃y ∈ Z, 6y − 3 = 28
(c) ∀p ∈ Q, ∃q ∈ Z, |p − q| ≤ 1
Recommended Problems
1. Which of the following are statements? If it is a statement, determine if it is true or false. No
justification is needed.
(a) 3 ≤ π
(b) 2x − 3 ≥ −1
(c) x2 − y 3 = 1
(d) N is a perfect square.
(e) x2 + 5x − 2
(f) x ≤ x + 1
(g) There is a largest real number.
(h) There is a smallest positive integer.
(i) Every real number is either positive or negative.
(j) Some triangles are right triangles.
2. For each of the following statements, identify the four parts of the quantified statement (quantifier,
variables, domain, and open sentence). Next, express the statement in symbolic form using as few
words as possible 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).
Finally determine if the original statement is true or false. No justification is needed.
(a) The equation x2 + 2x − 3 = 0 has a real solution.
(b) For all real numbers x and y, x 6= y implies that x2 + y 2 > 0.
(c) For every even integer a and odd integer b, a rational number c can always be found such that
a < c < b or b < c < a.
(d) There is a perfect square which is also a perfect cube.
3. Negate the following statements without using words or the ¬ symbol. For each statement determine
whether it or its negation is true.
(a) ∃a ∈ Z, ∀b ∈ Z, 3a = b
(b) ∀a ∈ Z, ∃b ∈ Z, 3a = b
(c) ∀a ∈ R, ∀b ∈ R, ∃c ∈ R,
a
=b
c
4. Express the following statement symbolically without using any words: Every integer is a perfect
square.
Mathematical Anecdote 1
R H Bing (1914–1986) was a topologist at the University of Texas, a student of R. L. Moore (1882–1974).
Many a Texas gentleman will forsake his first and middle names and use only his initials. Bing in fact
did not have a first and middle name. His full name, as recorded on his birth certificate, was R H Bing.
These were not initials, and there were no periods. One year Bing applies for a visa that he could take a
trip abroad. He filled out the application form, giving his name as usual. The form was returned to him,
and he was told that the State Department would not accept initials. The applicant must give complete
names. Bing wrote back, explaining patiently that his name was “ ‘R only’ ‘H only’ Bing.” In a few weeks
he received his visa in the mail. It was made out to “Ronly Honly Bing.”
R H Bing, 1914–1986
References: Steven G. Krantz, Mathematical Apocrypha: Stories and Anecdotes of Mathematics and the
Mathematical. The Mathematical Association of America, Inc., 2002.