What Does It Mean to Be Closed Under Addition
The Closure Property
Properties of Sets Nether an Functioning
Mathematicians are often interested in whether or not sure sets have particular properties under a given functioning. One reason that mathematicians were interested in this was and then that they could decide when equations would take solutions. If a ready under a given operation has sure general backdrop, then we can solve linear equations in that set up, for case.
There are several of import properties that a fix may or may not satisfy under a particular operation.� A property is a certain rule that holds if it is true for all elements of a set under the given operation and a property does not concord if there is at least one pair of elements that do non follow the property under the given operation.
Talking virtually backdrop in this abstract manner doesn't really brand whatever sense yet, and then let'south expect at some examples of properties and then that you lot can better understand what they are. In this lecture, we volition learn near the closure property.
The Property of Closure
A fix has the closure property under a item operation if the result of the performance is ever an chemical element in the fix.� If a set has the closure holding under a particular operation , then we say that the fix is � closed nether the operation .��
It is much easier to understand a property by looking at examples than it is by merely talking most information technology in an abstruse fashion, so permit's motility on to looking at examples so that you can see exactly what we are talking about when we say that a ready has the closure property:
Outset let�s look at a few infinite sets with operations that are already familiar to us:
a) The gear up of integers is airtight nether the operation of improver considering the sum of any two integers is always some other integer and is therefore in the set of integers.�
b) The set of integers is not closed nether the operation of partitioning because when y'all divide one integer by another, y'all don�t always get another integer as the answer.� For example, 4 and 9 are both integers, only 4 � 9 = 4/nine.� 4/nine is not an integer, so it is not in the set of integers!
At present permit�s expect at a few examples of finite sets with operations that may not exist familiar to u.s.a.:
e) The set {1,2,iii,4} is non airtight under the performance of addition because 2 + iii = 5, and 5 is not an element of the set {1,two,3,4}.�
We can meet this also by looking at the operation table for the fix {1,two,3,four} under the operation of addition:
| + | ane | ii | 3 | 4 |
| 1 | 2 | three | iv | 5 |
| 2 | 3 | 4 | 5 | 6 |
| 3 | 4 | 5 | half-dozen | 7 |
| iv | 5 | 6 | 7 | 8 |
The prepare{1,2,iii,iv} is not closed nether the operation + because there is at to the lowest degree one result (all the results are shaded in orange) which is non an element of the set up {ane,2,iii,iv}.� The chart contains the results 5, 6, 7, and 8, none of which are elements of the gear up {ane,two,3,4}!
f) The gear up { a,b,c,d,e } has the following functioning table for the performance *:
| * | a | b | c | d | e |
| a | b | c | due east | a | d |
| b | d | a | c | b | e |
| c | c | d | b | e | a |
| d | a | e | d | c | b |
| e | e | b | a | d | c |
The ready{ a,b,c,d,due east} is closed under the functioning * because all of the results (which are shaded in orangish) are elements in the set {a,b,c,d,e}.�
Now return to Blackboard to answer Group Lecture Questions two: Closure!
Source: http://www.cwladis.com/math100/Lecture2Groups.htm
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