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No, the size of infinities vary! Some are smaller than others.
Eg: the size of the set of numbers between 0 and 1 is infinite (0.1,0.11 and so on), but the size of this infinity is different from the size of the set of all numbers.
This is very very complicated maths and even the greatest mathematicians have avoided this paradox.
I don't know the proof well so just search around the net for it.
In mathematics, there are at least two categories of infinite sets - countably infinite, and uncountably infinite. Countably infinite means that you can set up a one to one correspondence between all members of the set and the set of natural number, aka "counting numbers".
Integers are countably infinite. You can pair them up with the natural numbers thus: 1,0 2,-1 3,1 4,-2 5,2 6,-3 and so on. The set of even numbers, odd number, and rational numbers are all countably infinite. The set of Real numbers, however is uncountably infinite. It can be shown that you can identify at least one real number that is not included in the set when you try to count all real numbers - thus while there are an infinite number of integers, there are even MORE real numbers. Not all infinite sets have the same number of elements.
What else can I help you with?
What is the relationship of number of elements to number of subset?
A finite set, with n elements has 2n subsets, including the empty set and itself. For infinite sets the number of subsets is the same order of infinity.
What are the finite or infinite sets?
A finite set is one containing a finite number of distinct elements. The elements can be put into a 1-to-1 relationship with a proper subset of counting numbers. An infinite set is one which contains an infinite number of elements.
When are two sets equivalent?
Two sets are considered equivalent when they contain the same number of elements, regardless of whether the elements themselves are the same or the order in which they are listed. This means there exists a one-to-one correspondence (bijective function) between the elements of the two sets. It’s important to note that equivalent sets can be of different types, such as finite and infinite sets, as long as their cardinalities match.
What are different kinds of cardinalities?
Cardinality refers to the number of elements in a set and can be categorized into several types: Finite Cardinality: Sets with a countable number of elements, such as the set of integers or the set of colors in a rainbow. Infinite Cardinality: Sets that have an unbounded number of elements, which can be further divided into countably infinite (like the set of natural numbers) and uncountably infinite (like the set of real numbers). Equal Cardinality: When two sets have the same number of elements, demonstrating a one-to-one correspondence between them. Understanding these types helps in set theory and various applications in mathematics and computer science.
What is the meaning of equivalent set?
Equivalent sets are sets with exactly the same number of elements.
What are kinds of sets according to number of elements what are kinds of sets according to number of elements?
Finite, countably infinite and uncountably infinite.
What is the cardinality of a union of two infinite sets?
The cardinality of finite sets are the number of elements included in them however, union of infinite sets can be different as it includes the matching of two different sets one by one and finding a solution by matching the same amount of elements in those sets.
What is the relationship of number of elements to number of subset?
A finite set, with n elements has 2n subsets, including the empty set and itself. For infinite sets the number of subsets is the same order of infinity.
What are the kinds of sets according to number of elements?
One possible classification is finite, countably infinite and uncountably infinite.
What are the finite or infinite sets?
A finite set is one containing a finite number of distinct elements. The elements can be put into a 1-to-1 relationship with a proper subset of counting numbers. An infinite set is one which contains an infinite number of elements.
When are two sets equivalent?
Two sets are considered equivalent when they contain the same number of elements, regardless of whether the elements themselves are the same or the order in which they are listed. This means there exists a one-to-one correspondence (bijective function) between the elements of the two sets. It’s important to note that equivalent sets can be of different types, such as finite and infinite sets, as long as their cardinalities match.
Two sets that contain the same number of elements are called what?
Two sets that contain the same number of elements are called "equinumerous" or "equipollent."
What is meaning of equivalent sets?
Equivalent sets are sets with exactly the same number of elements.
What is equal set from equivalent sets?
equal sets with exactly the same elements and number of elements.equivalent sets with numbers of elements
How do you determine if number sets same or different?
To determine if number sets are the same, compare their elements to see if they contain exactly the same numbers, regardless of order or repetition. If each number in one set can be matched to a number in the other set without any discrepancies, the sets are the same. If there are any differing elements or counts of elements, the sets are different. Using a method like sorting the sets or converting them to a list of unique elements can help in this comparison.
What is the meaning of equivalent set?
Equivalent sets are sets with exactly the same number of elements.
How do you describe the form of sets?
The form of sets is typically described by their elements, which are distinct objects considered as a whole. Sets can be represented using curly braces, such as {a, b, c}, where each letter represents an element. They can be finite, containing a limited number of elements, or infinite, containing an unbounded number of elements. Additionally, sets can be classified as subsets, universal sets, or power sets based on their relationships to other sets.
