Properties of Sets Under an Operation
Mathematicians are often interested in whether or not certain sets have particular properties under a given operation. One reason that mathematicians were interested in this was so that they could determine when equations would have solutions. If a set under a given operation has certain general properties, then we can solve linear equations in that set, for example.
There are several important properties that a set 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 hold if there is at least one pair of elements that do not follow the property under the given operation.
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. In this lecture, we will learn about the closure property.
The Property of Closure
A set has the closure property under a particular operation if the result of the operation is always an element in the set. If a set has the closure property under a particular operation, then we say that the set is “closed under the operation.”
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:
First let’s look at a few infinite sets with operations that are already familiar to us:
a) The set of integers is closed under the operation of addition because the sum of any two integers is always another integer and is therefore in the set of integers.
b) The set of integers is not closed under the operation of division because when you divide one integer by another, you don’t always get another integer as the answer. For example, 4 and 9 are both integers, but 4 ÷ 9 = 4/9. 4/9 is not an integer, so it is not in the set of integers!
to see more examples of infinite sets that do and do not satisfy the closure property.
c) The set of rational numbers is closed under the operation of multiplication, because the product of any two rational numbers will always be another rational number, and will therefore be in the set of rational numbers. 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. The only possible way that ac/bd could not be a fraction is if bd is equal to 0. But if a/b and c/d are both fractions, this means that neither b nor d is 0, so bd cannot be 0.
d) The set of natural numbers is not closed under the operation of subtraction because when you subtract one natural number from another, you don’t always get another natural number. For example, 5 and 16 are both natural numbers, but 5 – 16 = – 11. – 11 is not a natural number, so it is not in the set of natural numbers!
Now let’s look at a few examples of finite sets with operations that may not be familiar to us:
e) The set {1,2,3,4} is not closed under the operation of addition because 2 + 3 = 5, and 5 is not an element of the set {1,2,3,4}.
We can see this also by looking at the operation table for the set {1,2,3,4} under the operation of addition:
+ | 1 | 2 | 3 | 4 |
1 | 2 | 3 | 4 | 5 |
2 | 3 | 4 | 5 | 6 |
3 | 4 | 5 | 6 | 7 |
4 | 5 | 6 | 7 | 8 |
The set{1,2,3,4} is not closed under the operation + because there is at least one result (all the results are shaded in orange) which is not an element of the set {1,2,3,4}. The chart contains the results 5, 6, 7, and 8, none of which are elements of the set {1,2,3,4}!
f) The set {a,b,c,d,e} has the following operation table for the operation *:
a | b | c | d | e | |
a | b | c | e | 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 set{a,b,c,d,e} is closed under the operation * because all of the results (which are shaded in orange) are elements in the set {a,b,c,d,e}.
to see another example.
g) The set {a,b,c,d,e} has the following operation table for the operation $:
a | b | c | d | e | |
a | b | f | e | a | h |
b | d | a | c | h | e |
c | c | d | b | g | a |
d | g | e | d | c | b |
e | e | b | h | d | c |
The set{a,b,c,d,e} is not closed under the operation $ because there is at least one result (all the results are shaded in orange) which is not an element of the set {a,b,c,d,e}. For example, according to the chart, a$b=f. But f is not an element of {a,b,c,d,e}!
Now return to Blackboard to answer Group Lecture Questions 2: Closure!
