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Here is a correct proof by contradiction.
Assume that the natural numbers are bounded, then there exists a least upper bound in the real numbers, call it x, such that n ≤ x for all n.
Consider x - 1. Since x is the least upper bound, then x - 1 is not an upper bound; i.e. there exists a specific n such that x - 1 < n.
But then, x - 1 < n implies x < n + 1, hence x is not an upper bound.
QED
This concludes the proof; i.e. there exists no upper bound in the real numbers for the set of natural numbers.
P.S. There exists sets in which the set of natural numbers are bounded, but these are not in the real number system.
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Are the set of positive integers are equivalent to the set of natural numbers?
N-P = 0. Note here that N is the set of Natural numbers, P is the set of positive integers, and - is the set operation that in the above statement implies N and P are the same, with the exception that 0 is an element of N but not of P.
What is the set of natural numbers that are multiples of 8?
That's an infinite set that starts with 8, 16, 24, 32, 40 and goes on forever.
Is the LCM of a pair of numbers ever less than both numbers?
The LCM for any pair of natural numbers can be as big as their product.
What are the common factors for each set of numbers?
Please tell us your set of numbers.
What is the collective noun for numbers?
a set of numbers
which set show a subset of the set of natural numbers?
Natural numbers are just whole positive numbers. Since whole positive numbers can represent a distance along a line, they are a subset of real numbers.
What numbers are in the set of natural numbers and which are in the set of whole numbers?
All of the natural numbers.
What is difference between infinite Bounded and infinite unbounded set?
A set of numbers is bounded if there exist two numbers x and y (with x ≤ y)such that for every member of the set, x ≤ a ≤ y. A set is unbounded if one or both of x and y is infinite. Similar definitions apply for sets in more than 1 dimension.
What is the element of the intersection of the set of whole number and the set of natural numbers?
Well, honey, the intersection of the set of whole numbers and the set of natural numbers is the set of all positive integers. In other words, it's the numbers that are both whole and natural, which means it starts from 1 and goes on forever. So, there you have it, the sassy math lesson of the day!
Are natural numbers the same of rational numbers?
The set of rational numbers includes the set of natural numbers but they are not the same. All natural numbers are rational, not all rational numbers are natural.
What is the intersection between rational numbers and natural numbers?
It is the set of natural numbers.
Another name for the set of natural numbers?
Another name for a set of natural numbers is counting numbers.
Does the set of whole numbers is larger than the set of natural numbers?
If you mean larger by "the set of whole numbers strictly contains the set of natural numbers", then yes, but if you mean "the set of whole numbers has a larger cardinality (size) than the set of natural numbers", then no, they have the same size.
The set of numbers that include the natural numbers their opposites and 0?
The set of numbers that include the natural numbers, their opposites and 0 is called the set of integers.
What is a set of numbers including zero and all the counting numbers?
Whole numbers are the set of natural or counting numbers inclding zero
The set of whole numbers is equal to the set of natural numbers?
false, the set of natural numbers does not include 0, which can be considered a whole number.
How does the set of natural numbers differ from the set of whole numbers?
0 and negative integers are all whole numbers but they are not natural numbers.
