PROOFS: Use these theorems and others to prove these statements. Theorem 1: The sum of two...
PROOFS:
Use these theorems and others to prove these statements.
Theorem 1: The sum of two rational numbers is rational.
Theorem 2: The product of two rational numbers is rational.
Theorem 3: √ 2 is irrational.
Induction:
- Prove that 6 divides n 3 − n for any n ≥ 0
- Use strong induction to prove that every positive integer n can be written as the sum of distinct powers of 2. That is, prove that there exists a set of distinct integers, {k1, k2, . . . , km} where m ≥ 1, such that n = 2^(k1) + 2^(k2) + . . . + 2^(km).
Homework Answers
Add Answer to:
PROOFS:
Use these theorems and others to prove these statements.
Theorem 1: The sum of two...
For Exercises 1-15, prove or disprove the given statement. 1. The product of any three consecutive...
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 +...
Prove by strong induction that any nonzero natural number can be written as a sum of...
Prove by strong induction that any nonzero natural number can be written as a sum of distinct powers of 2.
please prove 9.6 and 9.7 The next three theorems formalize what you may have discovered in...
please prove 9.6 and 9.7
The next three theorems formalize what you may have discovered in the preceding group of questions. 9.6 Theorem. Let K be a positive integer Then, among any k real num- bers, there is a pair of them whose difference is within 1/K of being an integer When we take our collection of real numbers to be multiples of an ir- rational number, then we can find good rational approximations for the irrational number. Remember how...
Proofs Use the following definitions and facts about integers in writing your proofs. . Suppose n...
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...
Induction proofs. a. Prove by induction: n sum i^3 = [n^2][(n+1)^2]/4 i=1 Note: sum is intended...
Induction proofs. a. Prove by induction: n sum i^3 = [n^2][(n+1)^2]/4 i=1 Note: sum is intended to be the summation symbol, and ^ means what follows is an exponent b. Prove by induction: n^2 - n is even for any n >= 1 10 points 6) Given: T(1) = 2 T(N) = T(N-1) + 3, N>1 What would the value of T(10) be? 7) For the problem above, is there a formula I could use that could directly calculate T(N)?...
please prove the theorems, thank you very much 8.21 Theorem. A natural numbern can be written...
please prove the theorems,
thank you very much
8.21 Theorem. A natural numbern can be written as a sum of two squares of natural numbers if and only if every prime congruent to 3 modulo 4 in the unique prime factorization of n occurs to an even power Pythagorean triples revisited We are now in a position to describe the possible values for the hypotenuse in a primitive Pythagorean triple. 8.22 Theorem. If (a, h, e) is a primitive Pythagorean...
please prove lemma and theorems. 8.17 is not needed, thank you 8.15 Lemma. Let p be...
please prove lemma and
theorems. 8.17 is not needed, thank you
8.15 Lemma. Let p be a prime and let a be a natural number not divisible by p. Then there exist integers x and y such that ax y (mod p) with 0xl.lyl 8.16 Theorem. Let p be a prime such that p (mod 4). Thenp is equal to the sum of two squares of natural numbers. (Hinl: Iry applying the previous lemma to a square root of- mohulo...
Use strong induction to show that every positive integer can be written as a sum of...
Use strong induction to show that every positive integer can be written as a sum of distinct powers of two (i.e., 20 = 1; 21 = 2; 22 =4; 23 = 8; 24 = 16; :). For example: 19 = 16 + 2 + 1 = 2^4 + 2^1 + 2^0 Hint: For the inductive step, separately consider the case where k +1 is even and where it is odd. When it is even, note that (k + 1)=2 is...
answer question 5 please 3 and 4 are just included to refer to the theorems 3...
answer question 5 please 3 and 4 are just included to
refer to the theorems
3 Prove the following theorem: Theorem 2.2. Let S be a ser. The following statements are equivalent: (1) S is a countable set, i. e. there exists an injective function :S (2) Either S is the empty ser 6 or there exists a surjective function g: N (3) Either S is a finite set or there exists a bijective function h: N S (4) Prove...
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 Di...
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...
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 +...
Prove by strong induction that any nonzero natural number can be written as a sum of distinct powers of 2.
please prove 9.6 and 9.7
The next three theorems formalize what you may have discovered in the preceding group of questions. 9.6 Theorem. Let K be a positive integer Then, among any k real num- bers, there is a pair of them whose difference is within 1/K of being an integer When we take our collection of real numbers to be multiples of an ir- rational number, then we can find good rational approximations for the irrational number. Remember how...
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...
please prove the theorems,
thank you very much
8.21 Theorem. A natural numbern can be written as a sum of two squares of natural numbers if and only if every prime congruent to 3 modulo 4 in the unique prime factorization of n occurs to an even power Pythagorean triples revisited We are now in a position to describe the possible values for the hypotenuse in a primitive Pythagorean triple. 8.22 Theorem. If (a, h, e) is a primitive Pythagorean...
please prove lemma and
theorems. 8.17 is not needed, thank you
8.15 Lemma. Let p be a prime and let a be a natural number not divisible by p. Then there exist integers x and y such that ax y (mod p) with 0xl.lyl 8.16 Theorem. Let p be a prime such that p (mod 4). Thenp is equal to the sum of two squares of natural numbers. (Hinl: Iry applying the previous lemma to a square root of- mohulo...
answer question 5 please 3 and 4 are just included to
refer to the theorems
3 Prove the following theorem: Theorem 2.2. Let S be a ser. The following statements are equivalent: (1) S is a countable set, i. e. there exists an injective function :S (2) Either S is the empty ser 6 or there exists a surjective function g: N (3) Either S is a finite set or there exists a bijective function h: N S (4) Prove...
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...
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