Download Math 52 Practice Problems for Final Examination (4:30 pm 5/9/17

Math 52 Practice Problems for Final Examination (4:30 pm 5/9/17)
Exam problems will be similar to some of these, but the exam will be shorter
SETS
1a. Define what it means to say that S is a subset of T .
b. Use this definition to prove that S = {a ∈ Z : a ≡ 3 (mod 5)} is a subset of
T = {a ∈ Z : a2 ≡ 4 (mod 5)}
c. In (b), show that T is not a subset of S by exhibiting an element of T that is not an element of
S.
2. Suppose that S and W are subsets of T . Prove that S ∩ (S ∪ W ) = S
RELATIONS
3. In this problem, X = {1, 2, 3, 4} and P (X) is the collection of all subsets of X (called the “power
set” of X).
a. How many subsets does X have?
b. We define a relation R on S = P (X), i.e. on the subsets of X. So A and B will represent two
subsets of X. Remember that the number of elements in A is denoted by |A|. The relation is
R = {(A, B) : |A| = |B|}. So the subset A is related to the subset B if and only if A has the
same number of elements as B. Show that R is an equivalence relation.
c. Write out the equivalence class of A = {1, 3, 4}.
d. How many different equivalence classes are there?
e. How many different equivalence classes does the subset {2, 4} belong to?
f. Carefully state the main theorem about equivalence relations and partitions.
4. Two triangles T1 and T2 are similar if you can multiply the lengths of the sides of T1 by a positive
real number r and get the lengths of the sides of T2 .
a. Show that similarity is an equivalence relation on the set of all triangles.
b. Show that all equilateral triangles are in the same equivalence class for the equivalence relation
in part (a).
c. Show that there are infinitely many different equivalence classes of isosceles triangles.
5a. Let S = {2, 4, 6}. A binary relation R is defined on S by saying that s is related to t if and only if
s > t. Write R as a set of ordered pairs.
b. Is this relation reflexive?, symmetric? transitive?
INDUCTION
6a. Prove by induction that 3n < n! for all integers n ≥ 7.
6b. Suppose that a, b and m are positive integers with a ≡ b (mod m). Prove by induction that for each
n ∈ IN we have the congruence an ≡ bn (mod m).
7. A sequence is defined by a1 = 3, a2 = 3, and an+1 = an + 2an−1 for n ≥ 1. Prove by complete
induction that an = (2)n − (−1)n for all positive integers n.
8. THE EUCLIDEAN ALGORITHM
a. Describe the Extended Euclidean algorithm and what it is used for.
b. Suppose that a and b are two positive integers, and their greatest common divisor is g. Show
that if d is a positive integer such that d|a and d|b, then d|g.
c. Show how to use the Euclidean algorithm to solve for x and y such that 37x + 97y = 1.
MODULAR ARITHMETIC
9a. Use 8c) to find a number c such that 37c ≡ 1 (mod 97).
b. Let A = {1, 2, 3, . . . , 96}. A function f : A → A is given by f (k) = the remainder of 37k divided
by 97. Using the value found for c in part (d), a function g : A → A is given by g(j) = the
remainder of cj divided by 97. Show that f (k) ≡ 37k (mod 97) for all k in A and g(j) ≡ cj
(mod 97) for all j in A
10a.
11a.
12.
13a.
14a.
c. Show that g(f (k)) ≡ k (mod 97) for all k in A.
d. Show that g(f (k)) = k and f (g(j)) = j for all j in A.
e. Show that f is a bijection.
f. What is the remainder of 3796 divided by 97? Explain how this is related to part e.
How can you quickly find the remainder when a large positive integer is divided by 9?
b. Suppose that k is a large positive integer, and n has the same digits as k in a different order.
Show that k ≡ n (mod 9).
c. Show that k − n ≡ 0 (mod 9)
d. Show that the sum of the digits of k − n is a multiple of 9.
e. If k and n are three-digit natural numbers and the Great Sandzini knows that the middle digit
of k-n is a 9, how does he say what k − n is when someone just tells him the last digit?
If a is a positive integer which is not a multiple of 11, show that a10 ≡ 1 (mod 11).
b. If a is a positive integer which is not a multiple of 11, and n and m are any positive integers,
show that a10n+m ≡ am (mod 11).
c. What is the remainder of 926 divided by 11?
d. Find the remainder of 92689700000234900000087450293498345 divided by 11.
LOGIC
a. Define what it means for two logical statements to be equivalent.
b. If P and Q are two statements, show that the statement (¬P ) ∧ (P ∨ Q) is equivalent to the
statement Q ∧ ¬P .
c. Write the converse and the contrapositive of the statement “If you ace Math 52, then you
understand modular arithmetic and you understand equivalence relations.” Which of these is
equivalent to the original statement?
d. Write the negation of the following statement in a way that changes the quantifier: Every UVM
faculty member is a US citizen.
e. Write the negation of the following statement in a way that changes the quantifier: There exists
a differentiable function that is not continuous.
f. Write the negation of the following statement in a way that changes the quantifier: For all
integers n, the inequality n! > n2 holds.
g. Use a proof by contrapositive to prove that if c is an irrational number, then 1 + c is irrational.
h. Suppose that a, b and m are positive integers. Prove that a and b have the same remainder when
divided by m if and only if a = b + km for some integer k.
FUNCTIONS
According to our definition, what is a function?
b. Let A = {2, 4, 6} and B = {7, 8, 9, 10}. Suppose that we put F (2) = 8, F (4) = 7, and F (6) = 8.
Show that F is a function from A to B.
c. What is the definition of one-to-one?
d. Show that F is not one-to-one.
e. What is the definition of onto?
f. Show that F is not onto.
g. What is the range of F ? Write out this set.
h. Does F have an inverse function? Why or why not?
Suppose that f is a function from X to Y and g is a function from Y to Z. If f and g are onto, prove
that g◦f is onto.
b. If f and g have inverse functions, prove that g◦f has an inverse, namely f −1 ◦g −1 .
c. Find the inverse of the function h : R → R defined by h(x) = 2x5 + 10.