Final Exam Part 2
Ma 320-CRN 22189
Spring 2106
Carter
General Instructions There are two parts to this exam. In the first part, I will ask you
to define certain mathematical words and/or symbols. In the second part, you’ll use these
terms to prove theorems. Don’t worry! After you turn in the first part, I’ll give you a sheet
that reminds you of the definitions.
I am proud to have had you in my class, and I wish you the best for this test.
Don’t underestimate the satisfying nature of a piece of cheese melted over a slice of bread.
1. Recall the following:
(a) We say that d|n that is d divides n if d and n are integers and (∃q ∈ Z) such that
n = dq.
(b) The greatest common divisor gcd(a, b) of integers a, b is the largest number d =
gcd(a, b) such that d|a and d|b.
(c) A rational number is a number that can be written in the form q = a/b where
a, b ∈ Z, the denominator b 6= 0, and gcd(a, b) = 1.
(d) The union, ∪, of two sets A and B is the set A ∪ B = {x : x ∈ A or x ∈ B}.
(e) The intersection, ∩, of two set A and B is the set A ∩ B = {x : x ∈ A and x ∈ B}.
(f) A relation between sets X and Y is any non-empty subset R ⊂ X × Y = {(x, y) :
x ∈ X & y ∈ Y }.
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(g) A function Y ←− X is a relation R ⊂ X × Y such that for each x ∈ X there is a
unique y = f (x) ∈ Y with (x, f (x)) ∈ R.
f
(h) A function Y ←− X is injective or one-to-one if f (x1 ) = f (x2 ) implies that
x1 = x2 .
f
(i) A function Y ←− X is surjective or onto if (∀y)(∃x ∈ X) such that f (x) = y.
(j) The power set, P(X) of a set X consists of the set of all subsets of X.
(k) A relation on a set X (so X = Y ) is reflexive if (a, a) ∈ R (in in-fix notation
aRa).
(l) A relation on a set X (so X = Y ) is symmetric if (a, b) ∈ R, then (b, a) ∈ R. (in
in-fix notation aRb implies bRa).
(m) A relation on a set X (so X = Y ) is anti-symmetric if (a, b) ∈ R and (b, a) ∈ R,
then a = b. (in in-fix notation aRb & bRa implies a = b).
(n) A relation on a set X (so X = Y ) is transitive if (a, b) ∈ R and (b, c) ∈ R, then
(a, c) ∈ R. (in in-fix notation aRb & bRc implies aRc).
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2. Prove that
√
2 is an irrational number.
3. Suppose that A ⊂ B. Prove that P(A) ⊂ P(B).
4. If A has n elements prove that P(A) has 2n elements.
5. Prove that the composition of two injective (one-to-one) functions is injective.
6. Prove that the composition of two surjective (onto) functions is surjective.
7. Give an example of a pair of functions f and g such that f ◦ g 6= g ◦ f.
8. if a, b ∈ Z and m ∈ N define a ≡ b (mod m) if m|(b−a). Show that this is a symmetric,
reflexive, and transitive relation.
9. Show that d|n is a partial order on the positive integers. So the relation ·|· is reflexive,
anti-symmetric, and transitive.
10. Consider a set X = {1, 2, . . . , n} and a set Y = {1, 2, . . . , m}.
(a) What is the number of elements in the Cartesean product
X × Y = {(x, y) : x ∈ X & y ∈ Y }?
(b) In case n < m how many injective functions are there?
(c) Prove this.
11. Extra credit: prove the binomial theorem:
n
(x + y) =
n X
n
k=0
k
xn−k y k .
12. What is the most interesting thing that you learned this semester?
13. What is one thing that you wish you understood more?
14. Why do you think we should master the ability to prove or disprove mathematical
statements?
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