If A = {x : x is a natural number },               B = {x : x is an even natural number}  C = {x : x is an odd natural                                 number}                                             D = {x : x is a prime number }, find(i) A ∩ B (ii) A ∩ C (iii) A ∩ D(iv) B ∩ C (v) B ∩ D (vi) C ∩ D

Q: Let n>,2 and k be any boretine Proce that integers (n-1)* | n, 2

A: Given, n≥2 and k is any integer. n≡1modn-1Therefore, ni≡1modn-1 ∀i∈ℕ…

Q: If a,b, and c are consecutive integers and abc=0, then what is the smallest possible sum of there…

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Q: 1. If x is an odd integer, then x3 is odd.

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Q: Want a rigorous proof   Prove that The set of prime numbers has no largest element.

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Q: 3. Let z and y be 3-digit numbers such that y is equal to r written backwards. For example, if x =…

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Q: Show by contraposition that for all integers n^2, if  is odd, then n + 1 is even.

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Q: B) Prove that (O) = ) CE) n –k

A: We use the combination formula to simplify both sides

Q: If p + 2 is a prime number, prove that p² + 3 is divisible by 4.

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Q: For all integers x and y, if 6x+y is odd, then x and y are both odd. True False

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Q: 6. Use contradiction to prove that, for all integers k > 1, 2/k+1+ 2/K+2

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Q: Let a, b and c be integers. Prove that if a does not divide b and a|(b+c), then a does not divide c

A: Let a, b and c be integers.

Q: Prove that if for some integers a, b, c we have that a° + 6° + c° is divisible by 9, thên at least…

A: Here we have to prove that if for some integers a,b,c we have that a3+b3+c3 is divisible by 9,then…

Q: 5. Let a be a real number, c an integer, c > 0. Prove that

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Q: Prove: For any integer n, if n2 is divisible by 4, then n is even

A: I'm proving this using by contradiction.

Q: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4. Write P(0). Is P(0)…

A: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4. Write P(0). Is…

Q: Prove by using a direct proof that if n is an odd number, then n2 is also odd.

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Q: 3. Let a₁ = 2, a2 = 1, and for each natural number n > 2, let an 2an-1-an-2. Prove that for all…

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Q: Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the remainder…

A: we have to Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the…

Q: A. If p 2q25 is a prime number, prove that p-q is divisible by 3.

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Q: Use a proof by contrapositive to prove that, if (x + 12)/x is an integer, then x < 13.

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Q: Let A be the set of all positive integers whose last two digits form a number that is evenly…

A: We have the given condition as   Let us consider A be the set of all positive integers whose last…

Q: Suppose that J and K are points on the number line. If JK= 10 and K lies at 3, where could J be…

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Q: 7. Show that for every natural number n2 1, the integer ) + () +*(;) + - +) +22 +... +2" is…

A: Using binomial theorem:  n0+2n1+22n2....2nnn=n01n·20+n11n-1·21+n21n-2·22....nn1n-n·2n…

Q: Prove that for all integers a, b, and c, if a2 + b2 = c2, then at least one of a and b is even.

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Q: Prove A by Contradiction

A: Solution :-

Q: Show that if n is an integer and 5n + 6 is odd, then n is odd. (Use contradiction.)

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Q: For integer values of a, use a proof by contraposition (also called contrapositive or indirect…

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Q: integers a, b, and c with a≠0, if a∣b,  then a∣(bc). I need a know show table and a proof

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Q: Prove, directly from the definition of divisibility, that for any integer x#0, x|0, 1|x and x|x.

A: Use the following definition of divisibility to prove the required result. Let a and b be any two…

Q: 4. Let a, b, and d be integers such that dļab. Is it true that either da or d b? If you claim that…

A: Given integers a,b and d. We need to check whether if d/

Q: Let а, b and c be integers. Use the method of proof by contradiction to prove that if a|b and a f c,…

A: We have to solve given problem:

Q: Prove that N ≈ O*, where N is the set of natural numbers, and O* is the set of positive odd…

A: To prove:

Q: If n is an integer greater than 2, then for all integers m, n does not divide m or n + m # nm.

A: c Given n∈Zand n&gt;2. We have to prove that for all m∈Z, n/m or n+m/nm. Suppose n/m and…

Q: 6. Prove that if x is an irrational number and Y is a nonzero rational number, then xy is…

A: to prove this we take xy is rational number then show this is contradiction.

Q: An integer x is odd if x2 is odd. Let x, y and z be integers. If x y and y z, then x | z. For any…

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Q: A real number of the form p/2", where p is an integer and n is a nonnegative integer, is known as a…

A: We have to prove that there is a dyadic rational number between any two distinct real numbers. By…

Q: Prove that a natural number (an-1an-2a1ao)10 is divisible by 9 if and only if the sum of the digits…

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Q: Suppose that p is a prime number greater than 30. Prove that when p is divided by 30, the remainder…

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Q: 21. Prove that if n and k are natural numbers such that k < n, then (*:) - (-) - () n k k

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Q: Let a, b, and c be integers. Prove that if a divides b and a divides c, then a divides (b-c).

A: a, b and c are integers, a divides b and a divides c. Since a divides b, therefore b=ak,  k is any…

Q: for all integers a, b, and c, if a divides b and a divides c, then a divides (b2-c2). prove this…

A: Introduction: Divisibility is the base of number theory. For two integers a and b, if there exists…

Q: For any natural number n, n²+1 is always odd. True False

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Q: 3. Let z and y be 3-digit numbers such that y is equal to x written backwards. For example, if x =…

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Q: The Archimedean Property states that the set N of natural numbers is unbounded above in R. It is…

A: Given: The set ℕ of natural numbers is unbounded above in ℝ. To show: The following statements are…

Q: Suppose x is a real number and x is not equal to zero. Prove that if ((x^1/3)+5)/((x^2)+6)=1/x, then…

A: Given that, x is a nonzero real number, and 8 is not a root of given equation.

Q: Suppose x is a real number and x + 0. Prove that if then x 7 8. 2 16

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Q: Suppose we randomly select a subset of positive integers. How large must the subset be, if we want…

A: The Pigeonhole Principle:  If there are p+1 or more pigeons, need to distribute among p pigeonholes,…

Q: 4. Use a proof by contrapositive to prove that, for integers r and y: if a+y is odd, then exactly…

A: Here we use  Odd + Odd = Even Even +Even =Odd   Note:According to our guidelines we can solve only…

Q: A man finished a job in 5 days. On the first day, he finished 1/m of the job. On the second day, he…

A: GIVEN THAT JOB FINISHED = 5 DAYS

Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.