Loading...
End of Lesson
End of Lesson
Real numbers have certain behaviors when exposed to different operations. In Mathematics, sometimes we are dealing with subsets of Real Numbers and we define these subsets in different terms such as Groupoid, Semi-Group, Monoid, Group and others. These will be discussed in detail in Chapter 2. However, these classifications are defined by which properties can be applied on them.
Some of these propeties are closure, associative, commutative, distributive, identity, and inverse properties.
All mathematical properties are defined in terms of some operation. There are one, two, or more inputs and usually there is one output. Addition, subtraction, multiplication, division, union, intersection, complement, logarithm, exponentiation, negation, factorial, reciprocal, and absolute are all operations you might have heard about or used or will use in this course.
The properties we will be looking at here will be for binary operations.
Addition, subtraction, multiplication, division, union, intersection are all binary operations. Where two numbers interact with each other to produce a third number. Binary operations will be properly defined in chapter 2. They can be much more complex than simple addition or multiplication. For now, we will only be discussing the properties in the context of addition and multiplication as simple binary operations.
If the closure property is applied to a set for a specific operation, we say that "This set is closed w.r.t. (with respect to) this operation". Spoiler, The Real Number set is closed w.r.t. to both addition, multiplication and subtraction.
It means that applying an operation yields a number from the same set. If we add 2 and 3, both natural numbers, is the resulting number also a natural number? Yes. If this is true for all possible natural numbers such that you add any two natural numbrs and the result is always a natural number, then the natural number set is closed w.r.t. addition.
It means that applying an operation on members of a set yield a new number outside of the set. If we subtract 3 from 2, we get -1. -1 is not a natural number. So we say that the natural number set is not closed w.r.t. subtraction.
These statements are true, try and figure out why.
Read each statement carefully and decide if it is true or false.
Natural numbers are closed with respect to reciprocal.
Rational numbers are closed with respect to reciprocal.
Irrational numbers are closed with respect to multiplication.
Integers are closed with respect to addition.
Real numbers are closed with respect to addition.
Real numbers are closed with respect to multiplication.
This reads as, For all a and b in the Real Number set, a added to b will also be in the Real Number set.
This reads as, For all a and b in the Real Number set, a multiplied by b will also be in the Real Number set.
This reads as, For all a and b in the Real Number set, a operated on by b (or a asterisk b) will also be in the Real Number set.
This reads as, There exists some a and b in the Natural Number set, such that a minus b does not belong in the Natural Number set.
Closure under Addition of Natural Numbers
Closure under Multiplication for Real Numbers
Non-Closure under Subtraction for Natural Numbers
Closure under Subtraction for Real Numbers