Course: Theory of computation please answer the following questions using proof by construction, proof by contradiction and proof by induction
1) Show that the set of all integers is a countable set.
2) Show that mod 7 is an equivalence relation.
Course: Theory of computation please answer the following questions using proof by construction, proof by contradiction...
please answer questions #7-13
7. Use a direct proof to show every odd integer is the difference of two squares. [Hint: Find the difference of squares ofk+1 and k where k is a positive integer. Prove or disprove that the products of two irrational numbers is irrational. Use proof by contraposition to show that ifx ty 22 where x and y are real numbers then x 21ory 21 8. 9. 10. Prove that if n is an integer and 3n...
this needs to be proven using a direct proof, proof by
contraposition, or a proof by contradiction. YOU MUST SHOW ALL
STEPS LABEL THEM AND SPACE THEM WELL PLEASE. ALSO INCLUDE ALL
DEFINITIONS USED . FORMAL PROOFS ONLY PLEASE
5. 20 pts Prove that if n is an odd positive integer, then n 1 (mod 8)
Prove the following using proof by contradiction. Use a paragraph proof. GIF-<GIH Assume ΔGHF is NOT isosceles with FG t GH and also assume Prove that GI is not the median. (That is prove that F1 1. H1 ) Definition: A median in a triangle is a line segment that joins a vertex to the midpoint of the opposite side. 2. Assume ΔABC is isosceles. Prove that one of its base angles cannot be 95°.
From the proof of (ii) . Explain/Show why -n+ 1Sm-kn-1 is true by construction. . Explain/Show why 0 is the only number divisible by n in the range -n+1 ton-1 Proposition 6.24. Fix a modulus nEN. (i) is an equivalence relation on Z. (ii) The equivalence relation-has exactly n distinct equivalence classes, namely (ii) We need to prove that every integer falls into one of the equivalence classes [0], [1],..., [n -1], and that they are all distinct. For each...
In the following problem, we will work through a proof of an
important theorem of arithmetic. Your job will be to read the proof
carefully and answer some questions about the argument. Theorem
(The Division Algorithm). For any integer n ≥ 0, and for any
positive integer m, there exist integers d and r such that n = dm +
r and 0 ≤ r < m. Proof: (By strong induction on the variable n.)
Let m be an arbitrary...
1. What is wrong with the following proof that shows all integers are equal? (Please explain which step in this proof is incorrect and why is it so.) Let P(n) be the proposition that all the numbers in any set of size n are equal. 1) Base case: P(1) is clearly true. 2) Now assume that P(n) is true. That is for any set of size n all the numbers are the same. Consider any set of n + 1...
Please do both questions. wrong answers will be given thumbs
down.
Question 7. Prove using the Division Lemma that Yn E Z, n3 n is divisible by 3 (any proof not using the Division Lemma will receive no credit). Question 8. Define a relation ~ on R \ {0} by saying x ~ y İfzy > 0. (a) Prove that is an equivalence relation (b) Determine all distinct equivalence classes of~ prove that your answer is correct.
11: I can identify the predicate being used in a proof by mathematical induction and use it to set up a framework of assumptions and conclusions for an induction proof. Below are three statements that can be proven by induction. You do not need to prove these statements! For each one clearly state the predicate involved; state what you would need to prove in the base case; clearly state the induction hypothesis in terms of the language of the proposition...
please answer all the questions.
just rearranging. Explanation is not needed.
Use modular arithmetic to prove that 3|(221 – 1) for an integer n > 0. Hence, 3|(221 – 1) for n > 0. To show that 3|(221 – 1), we can show that (221 – 1) = 0 (mod 3). We have: (221 – 1) = (4” – 1) (mod 3) Then, (22n – 1) = (1 - 1) = 0 (mod 3) Since 4 = 1 (mod 3),...
all parts A-E please.
Problem 8.43. For sake of a contradiction, assume the interval (0,1) is countable. Then there exists a bijection f : N-> (0,1). For each n є N, its image under f is some number in (0, 1). Let f(n) :-0.aina2na3n , where ain 1s the first digit in the decimal form for the image of n, a2 is the second digit, and so on. If f (n) terminates after k digits, then our convention will be...