Prove by Contraposition: Vb e Z, if 362 – 2b is not divisible by some integer...
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...
Only need 2-5. Need it done ASAP, thank you in advance!! Proofs 1) (1.7.16) Prove that if m and n are integers and nm is even, then m is even or n is even. * What is the best approach here, direct proof, proof by contraposition, or proof by contradiction why? * Complete the proof. 2) Prove that for any integer n, n is divisible by 3 iff n2 is divisible by 3. Does your proof work for divisibility by...
Math 2300 - Homework 5 Due: Thursday, 10/10/2019 at 12:30 pm Notes: This homework consists of 6 problems. Write your answers in full sentences. Before you start a problem set up the goal and explain how will you proceed and why. Write all details thinking that the grader knows nothing about Mathematics. Pictures and graphs are great ideas to start a proof but will not be sufficient without written explanation. If you are referring to some result or theorem, please...
(2) For an integer n, let Z/nZ denote the set of equivalence classes [k) tez: k -é is divisible by n (a) Prove that the set Z/nZ has n elements. (b) Find a minimal set of representatives for these n elements. (c) Prove that the operation gives a well-defined addition on Z/nZ Hint: The operution should not depend on the choice of coset representatives Verify that this gives Z/n2 the structure of an ahelian group. Be sure to verify all...
Suppose a, b e Z. Show that if a for some integer c E Z, then a or b is even. Hint: Let P, A, and B be respectively the statements P {a2 + b2-c2}, A-fa is even), and B b is even]. In this problem you have to show that P (AVB). Use contradiction, i.e., prove that if the negation of this statement is true, then you come to a contradiction. Use that so that you have to assume...
Question 3 ONLY Notes: This homework consists of 3 problems. Write your answers in full sentences. Before you start a problem set up the goal and explain how will you proceed and why. Write all details thinking that the grader knows nothing about Mathematics. Pictures and graphs are great ideas to start a proof but will not be sufficient without written explanation. If you are referring to some result or theorem, clearly write its statement and location in the text....
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) л (PvQ) is equivalent to the statement Q^ P c. Write the converse and the contrapositive of the statement "If you earn an A in Math 52, then you understand modular arithmetic and you understand equivalence relations." Which of these d. Write the negation of the following statement in a way that changes the...
1. Prove that for any set S S R, S is closed if and only if Se is open. Notice the book has a proof of this, but it uses a different notation for set complements and a different definition of neighborhood. You may consult it, but you must write your proof using the definition for interior point I presented in class (also in the notes on blackboard). If you copy the proof from the book you will not receive...
Please show the solutions for all 4 parts! Problem 1 Let m E Z that is not the square of an integer (ie. mメ0, 1.4.9, ). Let α-Vm (so you have a失Q as mentioned above) (i) Prove the following:Qla aba: a,b Q is a subring of C, Za]a +ba: a, b E Z is a subring of Qla], and the fraction field of Z[a] is Q[a]. (3pts) (ii) Prove that Z[x]/(X2-m) Z[a] and Qx/(x2 mQ[a]. (3pts) i Let n be...
The Asc function in VB takes a character and return an integer that represents that character. It does not matter what the integer representing the character actually is, but what matters is this: Asc("a") is 1 less than Asc("b"), so that: x=Asc("a") thisLetter = x+1 # thisLetter is the Asc("b") This is a powerful fact that is used in encryption techniques, so data transferred over the web is 'deciphered so it is unreadable to others. To decipher data, we take...