1 point Prove the following statement: If n2 is even, then n is even. Order each of the following sentences so that the...
UUIDOR Quiz 2 - Ma Consider the following theorem. Theorem: The sum of any even integer and any odd integer is odd. Six of the sentences in the following scrambled list can be used to prove the theorem. By definition of even and odd, there are integers rands such that m = 2r and n = 2s + 1. By substitution and algebra, m + n = 2r + 25 + 1) = 2(r + s) + 1. Suppose m...
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...
#7. TRUE/FALSE. Determine the truth value of each sentence (no explanation required). ________(a) k in Z k2 + 9 = 0. ________(b) m, n in N, 5m 2n is in N. ________(c) x in R, if |x − 2| < 3, then |x| < 5. #8. For each statement, (i) write the statement in logical form with appropriate variables and quantifiers, (ii) write the negation in logical form, and (iii) write the negation in a clearly worded unambiguous English sentence....
1) Prove for any integer n, if n2 is a multiple of 6 then so is n. To get credit, you should use the following facts in your proof: If n2 is even then so is n. (Proved) If n2 is a multiple of 3, then so is n. (Proved) 2)By contradiction, prove that the square root of 6 is irrational. The result of part 1 should be be used as Lemma in your proof.
1. Consider the following claim. Claim: For two integers a and b, if a + b is odd then a is odd or b is odd. (a) If we consider the claim as the implication P =⇒ Q, which statement is P and which is Q? (b) Write the negations ¬P and ¬Q. (c) (1 point) Write the contrapositive of the claim. (d) Prove the contrapositive of the claim. 2. Use contraposition (proof by contrapositive )to prove the following claim....
Proofs Use the following definitions and facts about integers in writing your proofs. . Suppose n є Z. We say n is odd if there exists k є Z such that n-2k + 1 . Suppose n є Z. We say n is even if there exists ke Z such that n-2k . Suppose m, n є Z and m -0. We say ma divides n (written mln) if there exists k Z such that n mk. is either ever...
Prove that the following premise 4. Prove the following: (a) Prove that n is even if and only if n2 6n+5 is odd. (b) Prove that if 2n2 +3n +1 is even, then n is odd.
1.)Which of the expressions is equivalent to the following statement: The sum of two even numbers is even. a.) If x is even or y is even, then x + y b.) If x is even or y is even, then x + y is even c.) If x is even and y is even, then x + y is not even. d.) If x is even and y is even, then x + y is even 2.) Find a...
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...
For Exercises 1-15, prove or disprove the given statement. 1. The product of any three consecutive integers is even. 2. The sum of any three consecutive integers is even. 3. The product of an integer and its square is even. 4. The sum of an integer and its cube is even. 5. Any positive integer can be written as the sum of the squares of two integers. 6. For a positive integer 7. For every prime number n, n +...