1)prove that if x is rational and x not equal to 0, then 1/x is rational.
2) prove that there is a positive integers that equals the sum of the positive integers not exceeding it. Is your proof constructive or nonconstructive?
For 1) use the definition of a non-zero rational number.
Defn: A non-zero rational is any number that can be expressed as p/q where p and q are non-zero integers.
If p/q is rational, then how about q/p?
For 2) you should be able to supply both a constructive and non-consructive proof.
Since the sum of any finite set of integers is an integer, there exists an integer for the sum of the first +n integers.
There is a formula for this, but I'll let you work on this.
Constructive proof:
Let n be a positive integer. Then the sum of the first n positive integers is equal to n(n+1)/2. Therefore, for any positive integer n, there exists a positive integer that equals the sum of the positive integers not exceeding it.
Nonconstructive proof:
Let n be a positive integer. By the Well-Ordering Principle, there exists a positive integer that equals the sum of the positive integers not exceeding it.
For 1) To prove that if x is rational and x is not equal to 0, then 1/x is rational, we can use the definition of a non-zero rational number.
Let x be a rational number, so x can be expressed as p/q, where p and q are non-zero integers. Therefore, we have x = p/q.
We want to prove that 1/x is also a rational number. The reciprocal of x, denoted as 1/x, is equal to q/p.
To show that 1/x is rational, we need to demonstrate that q/p can be expressed as the ratio of two non-zero integers.
We can rewrite q/p as (q/1)/(p/1). Since both q and p are non-zero integers, q/1 and p/1 are non-zero rational numbers. The division of two non-zero rational numbers is still a rational number.
Therefore, we have shown that if x is a rational number and x is not equal to 0, then 1/x is also a rational number.
For 2) To prove that there is a positive integer that equals the sum of the positive integers not exceeding it, we can provide both a constructive and non-constructive proof.
Constructive Proof:
Let's assume that the positive integer we are looking for is n. According to the problem, n is the sum of all positive integers not exceeding it.
To find such an n, we can use the formula for the sum of an arithmetic sequence. The sum of the first n positive integers can be expressed as (n/2)(n+1).
Now, we set the equation (n/2)(n+1) = n. By simplifying and solving this equation, we can find the value for n, which would be our desired positive integer.
Non-Constructive Proof:
We can prove the existence of such a positive integer without explicitly finding its value.
Let S be the sum of all positive integers not exceeding a certain positive integer n. We assume that there is no positive integer that equals S.
Now, we consider the sum T = 1 + 2 + 3 + ... + n. This sum represents the total sum of all positive integers up to n.
Since S is the sum of all positive integers not exceeding n, it can be considered as a subset of T. Therefore, S is less than or equal to T.
However, according to the rules of set theory, the sum of any finite set of integers is an integer. Therefore, the sum T should be an integer.
But if there is no positive integer that equals S, it means that S is not an integer, which contradicts the fact that it is a subset of T.
Hence, our assumption that there is no positive integer equal to the sum of the positive integers not exceeding it must be incorrect.
Therefore, there must exist a positive integer that equals the sum of the positive integers not exceeding it.
1) To prove that if x is rational and x is not equal to 0, then 1/x is rational, we can use the definition of a non-zero rational number.
Let's assume that x is a rational number and x is not equal to 0. By the definition of a rational number, we can express x as p/q, where p and q are non-zero integers.
Now, let's consider 1/x. If we take the reciprocal of p/q, we get q/p. Since both q and p are non-zero integers, q/p can also be expressed as a rational number. Therefore, 1/x is rational.
We have proven that if x is rational and x is not equal to 0, then 1/x is rational.
2) To prove that there is a positive integer that equals the sum of the positive integers not exceeding it, we can provide both a constructive and a non-constructive proof.
Constructive proof:
We know that the sum of any finite set of integers is an integer. Therefore, for any positive integer n, the sum of the positive integers not exceeding n is an integer. We can express this as an equation: n = 1 + 2 + 3 + ... + n.
To find a specific positive integer that satisfies this equation, we can use the formula for the sum of an arithmetic series. The sum of the positive integers from 1 to n is given by the formula Sn = (n/2)(a + l), where a is the first term and l is the last term.
In this case, the first term a is 1 and the last term l is n. The sum Sn becomes n = (n/2)(1 + n). Simplifying this equation, we get n = (n^2 + n)/2, which can be rearranged to n^2 + n - 2n = 0. Factoring this equation, we have n(n+1)/2 = 0, which implies n(n+1) = 0.
From this, we can conclude that either n = 0 or n = -1. However, we are looking for a positive integer, so the only valid solution is n = 0. Therefore, 0 is the positive integer that equals the sum of the positive integers not exceeding it.
Non-constructive proof:
To prove the existence of a positive integer that satisfies the given condition without explicitly finding a value, we can use a non-constructive approach.
We know that the sum of any finite set of integers is an integer. Therefore, for any positive integer n, the sum of the positive integers not exceeding n is an integer.
Since we have established that the sum exists, we can conclude that there must be a positive integer that equals this sum. However, this non-constructive proof does not directly provide the value of the positive integer.
In summary, the proof for the existence of a positive integer that equals the sum of the positive integers not exceeding it can be both constructive, where we find a specific value, or non-constructive, where we establish the existence without explicitly determining the value.