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nicknaq
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Help me Understand "Closed Under Addition" and "Closed Under Multiplication"
Linear Algebra...matrices...etc
Examples would be great.
Thanks.
Linear Algebra...matrices...etc
Examples would be great.
Thanks.
When a set is closed under addition, it means that when you add any two elements from the set together, the result will also be an element in the set. In other words, the set contains all possible sums of its elements.
An example of a set that is closed under addition is the set of even numbers. When you add any two even numbers together, the result will always be an even number, which is also an element in the set.
Yes, a set can be closed under addition but not closed under multiplication. For example, the set of positive integers is closed under addition, but when you multiply two positive integers together, the result may not be a positive integer.
To prove that a set is closed under multiplication, you need to show that when you multiply any two elements in the set together, the result will also be an element in the set. This can be done using mathematical induction or by directly showing that all possible products are in the set.
It is important for a set to be closed under addition and multiplication because it allows for consistent and predictable results when performing mathematical operations on the set's elements. It also makes it easier to manipulate and solve equations involving the set's elements.