# Introduction to Abstract Mathematics

In-class Final Exam – Introduction to Abstract Mathematics
Show me your best math logic and math writing skills. This is the right moment to impress me (no pressure^¨ ). Attach the take-home, and do not forget
the portfolio. Be happy no matter what and enjoy summer the best you can!Your statement: The attached work is mine alone, and was completed without
notes, books, collaboration, or outside help. I understand I am not allowed tocopy or share this work with anybody else. My signature below represents my
understanding and acceptance of these conditions. Your online submission is yoursignature in Spring 2020.
Problem 1. For any sets A, B, C, D and E, show that
if A ⊆ B ∪ C, B ⊆ D and C ⊆ E then A ⊆ D ∪ E.
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Problem 2. Prove that
Xn
k=1
k · 3
k =
3
4
[(2n − 1) · 3
n + 1] , ∀n ∈ N.
Note that Pn
k=1 k · 3
k = 1 · 3 + 2 · 3
2 + · · · + n · 3
n
in case you don’t like the summation sign.
2
Problem 3. True or false? Prove your claim. For any integer n, if n3 + 5 is odd then n is even.
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Problem 4. (Find = educated guess. Proofs for extra credit.) I am overdoing this one a bit, because it is fun ¨^. There are, technically, four problems here.
(a) Find ∪An, ∩An for the family of sets An = {1, 2, 3, . . . , n}, n ∈ N. What happens if you replace the discrete sets An with the real intervals Bn = [1, n]?
(b) Find ∪Cα, ∩Cα for the family of intervals Cα = (−∞, α), α > 0. What happens if you replace the index set α > 0 with a new index set α < 0?
4
Problem 5. Let f : R \ {5} → R \ {2}, f(x) = 2x+3
x−5
.
(a) Prove that f is bijective.
(b) Determine f
−1
😕 →?, f(y) =?. You can use x instead of y if it is easier for you.
(c) Double check your rule for f
−1 by computing f ◦ f
−1 and f
−1 ◦ f. Draw conclusions.
5
Problem 6. Let f : A = {1, 2, 3, 4} → B. Describe a codomain B and a rule for f (which could be a clear bubble diagram if you want) such that each of the following conditions hold individually. In each case find f(A) and f({1, 2}).
(a) f is onto but not one-to-one
(b) f is not onto but it is one-to-one
(c) f is onto and one-to-one
(d) f is neither onto nor one-to-one
6
Problem 7. For each natural number n, let Tn = {x ∈ N | x > n}. Consider the following statement: (∀a ∈ N)(∃n ∈ N)(a ∈ Tn ∨ a ≤ 1).
(a) Write a negation for the statement above.
(b) Prove or disprove (a).
7
Problem 8. Select your favorite proof problem from this semester and write it here with a detailed proof. Explain your choice. (You are welcome to include multiple problems if you cannot choose only one. I am sure you have several favorites ¨^!)
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Extra credit for extra fun: these points are added to the total number of points from the previous problems and averaged.
♠ (4 points) Complete the following statement so that it is true and provide a proof: Let
a, b ∈ R. Then √
a + b =

a +

b if and only if ………….
♣ (4 points) Assign a grade of A (correct), C (partially correct), or F (failure) to the following proof. Justify an assignment of grade other than A.
Claim: The number 25 cannot be written as the sum of three integers, an even number of which are odd.
Proof: Assume, to the contrary, that 25 can be written as the sum of three integers, an even number of which are odd. Then 25 = x + y + z, where x, y, z ∈ Z. We consider two cases.
Case 1. x and y are odd. Then x = 2a + 1, y = 2b + 1, z = 2c, where a, b, c ∈ Z.
Therefore 25 = x + y + z = (2a + 1) + (2b + 1) + 2c = 2(a + b + c + 1). Since a + b + c is an integer, 25 is even, a contradiction.
Case 2. x, y, and z are even. Then x = 2a, y = 2b, z = 2c, where a, b, c ∈ Z. Hence
25 = x + y + z = 2a + 2b + 2c = 2(a + b + c). Since a + b + c is an integer, 25 is even, again a contradiction.
Q.E.D.
♦ (4 points) The standard definition of a real function f being continuous at a point x = a
is (∀ > 0)(∃δ > 0)(∀x)[|x − a| < δ ⇒ |f(x) − f(a)| < ]. Write down the formal definition
for f being discontinuous at a.
♥ (4 points) Construct a bijection f : N → Z if possible; if not possible explain why not.

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