set theory and logic 8. (5 points) Let r be an integer. Write a proof by...
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...
2. Using the given logic rules, write a proof sequence for the following assertion. (11 points) pva 4 → pist st זר 74 → Us Statements 1. p Va 2.91 3.(p Ast 4. r 5.9 → (UA) 6. Reasons Given Given Given Given Given 7. 8. 9. 10. 11. t
Problem (2), 10 points Let n be an integer. Prove that if 3 does not divide n, then 3(2n2 5) Problem (2), 10 points Let n be an integer. Prove that if 3 does not divide n, then 3(2n2 5)
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)
3. Let the relation R be defined on the set R by a Rb if a -b is an integer. Is R and equivalence relation? If yes, provide a proof. Consider the equivalence relation in #3. a. What is the equivalence class of 3 for this relation? 1 b. What is the equivalence class of for this relation? 2
3. Let W = P({1,2,3,4,5}). Consider the following statement and attempted proof: VAE W WB EW (((AUB) C A) + (ACB)) (1) Towards a universal generalization argument, choose arbitrary A € W, BEW. (2) We need to show ((AUB) C A) + (ACB). (3) Towards a proof by contraposition, assume B CA, and we need to show A C (AUB). (4) By definition of subset inclusion, this means we need to show Vc (E A →r (AUB)). (5) Towards a...
Question 1. Let x be an integer. Show that if r2 – 4.+ 17 is odd, then x is even. Make sure to show all steps and indicate the type of proof used. (9 points)
proof should involve r^(2i) for some i Adventures in Algebra VII: This is completely normal. 1 Let n be a positive even integer. Recall that the dihedral group D is generated by r ands subject to r" = s? = c and rs = sr-1. Show that (-2) is a normal subgroup of Dm
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.
(i) Show that R is a subring of the polynomial ring Rx. | R{]4 (ii) Let k be a fixed positive integer and be the set of all polynomials of degree less than or equal to k. Is R[xk a subring of R[a]? 2r4+3x - 5 when it is (iii) Find the quotient q(x divided by P2(x) of the polynomial P1( and remainder r(x) - 2c + 1 in - (iv) List all the polynomials of degree 3 in Z...