MATH 2200 Quiz #1
Thomas Goller
September 11, 2012
Instructions:
• Write your responses to the following questions on the scratch paper provided. Don’t turn in this sheet!
• Label your responses with the question numbers.
• Pick and choose those questions (or parts of questions) that you can do most easily!
• Please write clearly!
• Look at the back of this sheet!
• Write your name at the top of the first sheet of your written solutions. Staple the pages of your written
solutions as you turn them in!
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1.1
Logic
Truth tables
Show that P =⇒ Q and ¬P ∨ Q are logically equivalent using truth tables. Use this to prove (without
truth tables) that (P ∧ Q) =⇒ R is logically equivalent to P =⇒ (Q =⇒ R) (hint: show that both of
these compound propositions are logically equivalent to R ∨ ¬P ∨ ¬Q).
1.2
Arguments and validity
What is an argument? What does it mean for an argument to be valid ? Give an example of a valid argument
with conclusion “Horses are insects”. Give an example of an invalid argument. Can an argument with a
true conclusion be invalid?
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2.1
Proofs Using Integers
Evenness and oddness
Define what it means for an integer to be even. Define what it means for an integer to be odd. State and
prove a theorem about the integers pertaining to evenness or oddness.
2.2
Types of proof
What are the three main types of proofs? Explain (possibly by giving examples) why each type of proof can
be particularly useful.
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3
3.1
Functions
Injectivity and surjectivity
What is a function? What do we call the most important sets associated to a function? What does it mean
for functions to be injective or surjective? Give an example of an injective function that is not surjective,
and a surjective function that is not injective. Give an example of a function that is neither injective nor
surjective, and a function that is both injective and surjective (bijective).
3.2
Inverses
What condition on a function guarantees that it has an inverse? What is the composition condition that
a function and its inverse must satisfy? Give an example of a function and its inverse, and check that the
composition condition holds.
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Challenge
Prove the following theorem:
Theorem. For n ≥ 3, there are no positive integers x, y, z such that xn + y n = z n .
Give a counterexample to show the theorem is false for n = 2.
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