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Evgeny.Makarov
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Call a nonempty (finite or infinite) set $A\subseteq\Bbb R$ complete if for all $a,b\in\Bbb R$ such that $a+b\in A$ it is also the case that $ab\in A$. Find all complete sets.
[sp]Since $A$ is nonempty it contains some real number $a$. Then $0+a = a\in a$, so the completeness condition implies that $0 = 0a\in A.$Evgeny.Makarov said:Call a nonempty (finite or infinite) set $A\subseteq\Bbb R$ complete if for all $a,b\in\Bbb R$ such that $a+b\in A$ it is also the case that $ab\in A$. Find all complete sets.
Complete sets of real numbers refer to a collection of numbers that includes all the possible values between two given numbers. This set includes both rational and irrational numbers.
To find all the numbers in a complete set of real numbers, you can use a number line or a mathematical equation. For example, to find all the numbers between 1 and 5, you can list them out as 1.1, 1.2, 1.3, and so on until you reach 4.9.
No, there are no missing numbers in a complete set of real numbers. This set includes all the possible values between two given numbers, so there are no numbers left out.
The main difference between a complete set of real numbers and an incomplete set of real numbers is that a complete set includes all the possible values between two given numbers while an incomplete set only includes some of the values.
Complete sets of real numbers are important in mathematics because they allow for precise and accurate calculations. They also help in understanding the properties and relationships between different numbers and their operations.