Show that for any square-free integer n > 1, √ n is an irrational number

Expert Solution

Colby Messinger answered 1 year ago

1. Use cardinality to show that between any two rational numbers there is an irrational number....

1. Use cardinality to show that between any two rational numbers there is an irrational number. Hint: Given rational numbers a < b, first show that [a,b] is uncountable. Now use a proof by contradiction. 2. Let X be any set. Show that X and P(X) do not have the same cardinality. Here P(X) denote the power set of X. Hint: Use a proof by contradiction. If a bijection:X→P(X)exists, use it to construct a set Y ∈P(X) for which Y...

Determine the number of permutations of {1,2,3,...,n-1,n} where n is any positive integer and no even...

Determine the number of permutations of {1,2,3,...,n-1,n} where n is any positive integer and no even integer is in its natural position.

For all integers n > 2, show that the number of integer partitions of n in...

For all integers n > 2, show that the number of integer partitions of n in which each part is greater than one is given by p(n)-p(n-1), where p(n) is the number of integer partitions of n.

Use cardinality to show that between any two rational numbers there is an irrational number. Hint:...

Use cardinality to show that between any two rational numbers there is an irrational number. Hint: Given rational numbers a < b, first show that [a, b] is uncountable. Now use a proof by contradiction

Partitions Show that the number of partitions of an integer n into summands of even size...

Partitions Show that the number of partitions of an integer n into summands of even size is equal to the number of partitions into summands such that each summand occurs an even number of times.

4. Use a proof by contradiction to show that the square root of 3 is irrational....

4. Use a proof by contradiction to show that the square root of 3 is irrational. You may use the following fact: For any integer k, if k2 is a multiple of 3, then k is a multiple of 3. Hint: The proof is very similar to the proof that √2 is irrational. 5. Use a direct proof to show that the product of a rational number and an integer must be a rational number. 6. Use a proof by...

Is rational number divided by an irrational number equal to irrational number or rational number?

Is rational number divided by an irrational number equal to irrational number or rational number? for example such as ( 5 / 2pi )

For any nonnegative integer n, let Y1 < Y2 < · · · < Y2n+1 be...

For any nonnegative integer n, let Y1 < Y2 < · · · < Y2n+1 be the ordered statistics of 2n + 1 independent observations from the uniform distribution on [−2, 2]. (i) Find the p.d.f. for Y1 and the Y2n+1. (ii) Calculate E(Yn+1). Use your intuition to justify your answer.

Consider an n×n square board, where n is a fixed even positive integer. The board is...

Consider an n×n square board, where n is a fixed even positive integer. The board is divided into n 2 unit squares. We say that two different squares on the board are adjacent if they have a common side. N unit squares on the board are marked in such a way that every unmarked square on the board is adjacent to at least one marked square. Determine the smallest possible value of N.

Let E = Q(√a), where a is an integer that is not a perfect square. Show...

Let E = Q(√a), where a is an integer that is not a perfect square. Show that E/Q is normal

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