But if a/b and c/d are both fractions, this means that neither b nor d is 0, so bd cannot be 0.ĭ) The set of natural numbers is not closed under the operation The only possible way that ac/bd could not be a fraction is if bd is equal to 0. This is because multiplying two fractions will always give you another fraction as a result, since the product of two fractions a/b and c/d, will give you ac/bd as a result. Rational number, and will therefore be in the set of rational numbers. To see more examples of infinite sets that do and do not satisfy theĬ) The set of rational numbers is closed under the operation of multiplication,īecause the product of any two rational numbers will always be another Is not an integer, so it is not in the set of integers! For example, 4 and 9 are both integers, but 4 ÷ 9 = 4/9. When you divide one integer by another, you don’t always get another integerĪs the answer. The sum of any two integers is always another integer and is thereforeī) The set of integers is not closed under the operation of division because It is much easier to understand a property by looking at examples than it is by simply talking about it in an abstract way, so let's move on to looking at examples so that you can see exactly what we are talking about when we say that a set has the closure property:įirst let’s look at a few infinite sets with operations thatĪ) The set of integers is closed under the operation of addition because That the set is “ closed under the operation.” IfĪ set has the closure property under a particular operation, then we say Operation if the result of the operation is always an element in the set. The Property of ClosureĪ set has the closure property under a particular In this lecture, we will learn about the closure property. Talking about properties in this abstract way doesn't really make any sense yet, so let’s look at some examples of properties so that you can better understand what they are. It is true for all elements of a set under the given operation andĪ property does not hold if there is at least one pair of elements thatĭo not follow the property under the given operation. A property is a certain rule that holds if May or may not satisfy under a particular operation.
There are several important properties that a set
If a set under a given operation has certain general properties, then we can solve linear equations in that set, for example. One reason that mathematicians were interested in this was so that they could determine when equations would have solutions. Mathematicians are often interested in whether or not certain sets have particular properties under a given operation.
The Closure Property Properties of Sets Under an Operation
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