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Question 1 We prove 0x = 0 as below. Which method of proof did we use? X=X X-x = 0 (1-1)x =0 0x =0 direct proof proof by cases proof by contrapositive Question 2 If direct proof is used to prove the following statement: If x is a real number and x s 3, then 12 - 7x + x*x > 0. What is the hypothesis? 12- 7x+x*x>0 If x is a real number and xs 3 12-7x+x*x<0 If x is not a real number or x > 3 Question 3 If proof by contrapositive is used...
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ntifiers , Counterexamples, Disproof (#9, 15
pts)
#9. For each statement, state whether the statement is
true or false. If false, explain; provide a counterexample
as appropriate or a careful explanation. (If true, no
explanation expected)
(a) n in N, n+23 ≥n3+8.
(b) x in R, x+23 ≥x3+8.
(c) n in N, 4n + 1 is prime.
(d) x, y in R, if |x|
< |y|, then x2 < xy.
(e) m in N such that n in
N, m...
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#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....
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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...
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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...
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Let x,y ∈ R. Which of the following statements are true. If the
statement is true prove it, if not give a counterexample
a) If x is rational and y is irrational, then x y is irrational. b) If x and y are both irrational then x + y is irrational. c) Ifx and y are both irrational then ry is irrational d) Ifx is rational and y is irrational then ry is irrational.
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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 +...
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Proof by contradiction that the product of any nonzero rational number and any irrational number is irrational (Must use the method of contradiction). Which of the following options shows an accurate start of the proof. Proof. Let X+0 and y be two real numbers such that their product xy=- is a rational number where c, d are integers with d 0. Proof. Let x0 and y be two real numbers such that their product xy is an irrational number (that...
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Please help answer all parts!
(1) Prove that 75 is irrational. (State the Lemma that you will need in the proof. You do not need to prove the lemma.) (2) Disprove: The product of any rational number and any irrational number is irrational. (3) Fix the following statement so that it is true and prove it: The product of any rational number and any irrational number is irrational. (4) Prove that there is not a smallest real number greater than...
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1 point Prove the following statement: If n2 is even, then n is even. Order each of the following sentences so that they form a logical proof. Proof by Contrapositive: Choose from these sentences: Your Proof: Suppose n is odd. Then by definitionn 2k +1 for some integer k Required to show if n is not even (odd), then n is not even (odd). Thus n2(2k1)2. n24k2 4k1. 22(22+2k) +1 Thus n2 (an integer) +1 and by definition is odd....