mathematical proofs
exam
mathematical proofs代写 Question A. Each question in this section is worth two marks for a total of 40 marks.Each question in this section has only one ···
Question A. [40 marks] mathematical proofs代写
Each question in this section is worth two marks for a total of 40 marks.
Each question in this section has only one correct answer.
Your answers must be transferred to the table provided on the fifinal page to be evaluated.
Not doing so will result in a zero grade for this question.
Part 1 [2 marks]
Suppose that p(x) is a quadratic polynomial, and that s(x) = p(x + 2), for all x 2 R.
A. p(x) and s(x) have the same discriminant.
B. The discriminant of p(x) is 2 more than the discriminant of s(x).
C. The discriminant of s(x) is 2 more than the discriminant of p(x).
D. The discriminant of s(x) and the discriminant of p(x) are di↵erent, but by more than 2.
E. None of A., B., C., or D.
Part 2 [2 marks]
Is the following statement true or false?
(∀x∈ R)(∃y∈ R) [y ≤ x ⇒ 0 ≤ x]
A. True, because we can take y = −1.
B. True, because we can take y = x.
C. True, because we can take y = x².
D. True, because we can take x = 1.
E. None of A., B., C., or D.
Part 3 [2 marks]
Which of the following statements is logically equivalent to P⇔ Q?
A. P∧ Q
B. P ∨ Q
C. (P⇒ Q) ∧ (¬P ⇒ ¬Q)
D. (P⇒ Q) ∧ (¬Q ⇒ ¬P)
E. None of A., B., C., or D.
Part 4 [2 marks] mathematical proofs代写
Mike conjectures that for all integers a, b: “If ab is a prime, then a is prime or b is prime.” Which of the following strategies can be used in a proof of this statement?
A. Assume that a and b are both not prime. Prove that ab is prime.
B. Assume that ab is prime, and prove that both a and b are not prime.
C. Assume that ab is prime. Derive a contradiction.
D. Assume that ab is prime and that both a and b are not prime. Derive a contradiction.
E. None of A., B., C., or D.
Part 5 [2 marks]
Which of the following sets is not empty?
A. ∅
B. {x : x ∈ {∅}}
C. N x ∅ ;
D. N \ Z
E. None of A., B., C., or D.
Part 6 [2 marks]
Consider the function f : {0, 1} → R defifined by f(x) = x². Consider the function g : {0, 1} → R defifined by g(x) = x. Are these functions equal?
A. No, because they have di↵erent codomains.
B. No, because they have di↵erent ranges.
C. No, because they have di↵erent defifinitions; x ≠ x².
D. Yes.
E. None of A., B., C., or D.
Part 7 [2 marks]
Consider the relation ≤ on R.
A. ≤ is not reflflexive.
B. ≤ is not symmetric.
C. ≤ is not transitive.
D. ≤ is an equivalence relation.
E. None of A., B., C., or D.
Part 8 [2 marks] mathematical proofs代写
What is 20192+ 20194 + 20196 + 20198 congruent to modulo 4?
A. 0
B. 1
C. 2
D. 3
E. None of A., B., C., or D.
Part 9 [2 marks]
Let f : A → B be a function, and let C ∈ A. Consider the conjecture:
f(A \ C) = f(A) \ f(C).
Is this conjecture true or false?
A. True, because if A = C then the conjecture becomes ∅ = ∅ which is true.
B. True, by showing that each set is a subset of the other.
C. False. For example f : R → R defifined by f(x) = |x| and C = {0}.
D. False. For example f : R → R defifined by f(x) = |x| and C = [0 , ∞).
E. None of A., B., C., or D.
Part 10 [2 marks]
Suppose that −2 <x< 1 and −3 <y< 4. Which of the following is true?
A. −7 < |2x + y| < 6
B. 0 < |2x + y| < 6
C. 0 ≤ |2x + y| < 6
D. 0 ≤ |2x + y| ≤ 7
E. None of A., B., C., or D.
Part 13 [2 marks]
Suppose that f : N → {0, 1} is a function, and Yoshio knows that:
1.f(2) = 1,
2.f(3) = 1, and
3.∀n ∈ N [f(n)=1 ⇒ f(n + 2) = 1].
What is the set of all n such that Yoshio knows that f(n) = 1?
A. All ∈ 2 N with n ≥ 3.
B. All ∈ 2 N with n ≥ 2.
C. All ∈ 2 N.
D. All even natural numbers.
E. None of A., B., C., or D.
Part 14 [2 marks]
Defifine a sequence by a1 = 1, a2 = 2 and an+2 = an+1 · an for n 2 N. What is a10?
A. 210
B. 234
C. 221
D. 29
E. None of A., B., C., or D.
Part 15 [2 marks]
Let f : A → B and g : B → C be functions. Suppose that g ◦ f is a surjection. Which of the following must be true?
A. g is a surjection.
B. f is a surjection.
C. g is a bijection.
D. f ◦ g is a surjection.
E. None of A., B., C., or D.
Part 16 [2 marks] mathematical proofs代写
Which of the following is NOT equivalent to f : A → B is an injection?
A. ∀x1, x2 ∈ A, if x1 = x2, then f(x1) = f(x2).
B. ∀x1, x2 ∈ A, if x1 ≠ x2, then f(x1) ≠ f(x2).
C. ∀x1, x2 ∈ A, if f(x1) = f(x2), then x1 = x2.
D. For all b∈ B there is at most one a∈ A such that f(a) = b.
E. None of A., B., C., or D.
Part 17 [2 marks]
Let A, B be sets. Which of the following is NOT equivalent to |A| ≤ |B|?
A. There is an injection f : A → B.
B. There is a set C ⊆ B and a injection g : A → C.
C. There is a set C ⊆ B and a bijection h : A → C.
D. A ⊆ B.
E. None of A., B., C., or D.
Part 18 [2 marks]
Which of the following sets is infifinite and countable? (Here, P(A) is the power set of A.)
A. P(N)
B. Q x Z
C. P({1, 2, 3,…, 2019})
D. P(R)
E. None of A., B., C., or D.
Part 19 [2 marks]
How many divisors does 1 · 2 · 3 · 4 have?
A. 4
B. 6
C. 8
D. 24
E. None of A., B., C., or D.
Part 20 [2 marks] mathematical proofs代写
Compute gcd(21325374, 24335271).
A. 2 · 3 · 5 · 7
B. 25·35·55·75
C. 24·33·53·74
D. 21·32·52·71
E. None of A., B., C., or D.
Long Answer mathematical proofs代写
The following six questions are worth fifive marks each for a total of 30 marks.
Show your work. Unsupported solutions will receive little or no credit.
Question B. [5 marks]
Part 1 [3 marks]
Let A, B be sets. Prove that if A ∩ B = ∅, then (A x B) ∩ (B x A) = ∅.
Part 2 [2 marks]
Is the converse of Part 1 true? If it is then give a proof. If it isn’t then provide a counterexample.
Question C. [5 marks]
Let x, y > 0 be positive real numbers. Prove that
Question D. [5 marks]
Part 1 [2 marks]
Negate the statement:
(∀x ∈ R)(∃y ∈ R) [y ≤ x ⇒ 0 ≤ x] .
Part 2 [3 marks]
Prove that √6 is irrational.
Question E. [5 marks] mathematical proofs代写
Part 1 [3 marks]
Let E1, E2 be equivalence relations on N. Prove that |E1| = |E2|.
Part 2 [2 marks]
George conjectures that “For all sets A, if E1, E2 are equivalence relations on A, then |E1| = |E2|.”Show that his conjecture is false by providing a counterexample. (In this part, if something is an equivalence relation, you may state it without proof.)
Question F. [5 marks] mathematical proofs代写
Part 1 [3 marks]
Apply the Euclidean algorithm to a = 21 and b = 13 to fifind gcd(13, 21).
Part 2 [2 marks]
Find integers x, y such that 21x + 13y = 1.
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