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Problem: Mrs. Glosser inquiry Kyesha, Angie and also Eduardo to list the set all that integers better than -3.

You are watching: All real numbers that are less than -3 or greater than or equal to 5

Analysis: Each student created this collection using various notation.

Solution:

KyeshaP = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
AngieP = -2, -1, 0, +1, +2, +3, +4, +5, +6, +7, +8, +9, +10, +11, ...
EduardoP = all integers greater than -3

Each the the students in the problem above used correct notation! However, Mrs. Glosser said them the there to be another way to write this set:

P = x : x is one integer, x > -3 , i m sorry is review as: “P is the set of elements x such that x is one integer better than -3.”

Mrs. Glosser used set-builder notation, a shorthand provided to create sets, regularly sets through an infinite number the elements. Let"s look in ~ some an ext examples.

ExampleSet-Builder NotationRead asMeaning
1x : x > 0the collection of all x such that x is greater than 0.any value better than 0
2x : x ≠ 11the set of all x such that x is any number other than 11.any value other than 11
3{x : x 

Each of these sets is check out aloud precisely the same way when the colon : is changed by a upright line | as in  x > 0. Both the colon and the upright line stand for the indigenous "such that". Let"s look at these examples again.

ExampleSet-Builder NotationRead asMeaning
 with :with |  
1x : x > 0 x > 0the collection of all x such that x is greater than 0any value better than 0
2x : x ≠ 11 x ≠ 11the set of all x such that x is any kind of number except 11any value except 11
3{x : x x | x 

Note that the "x" is just a place-holder, it can be anything, such as  q .

The general type of set-builder notation is:

General Form: formula for elements : restrictions or  restrictions

Types of Numbers

In the examples above, us examined values v set-builder notation. However, we did no specify what form of number these values can be. With set-builder notation, us normally show what form of number we room using. Because that example, watch at x below:

 x > 3

Recall that  means "a member of", or merely "in".  is the one-of-a-kind symbol for Real Numbers. So x means "all x in ".

Thus, x  means "the collection of all x in  such that x is any kind of number higher than 3." (In other words, x is all real numbers greater than 3.)

There are other species of number besides genuine Numbers. Right here are some common types used in mathematics.

Common types of Numbers
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Natural NumbersIntegersRational NumbersReal NumbersImaginary NumbersComplex Numbers

Whole Numbers start at zero and also go up by one forever (no fractions). The collection of entirety numbers is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...

Counting Numbers are totality numbers greater than zero. (You can not count with zero!) The set of counting number is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...

Natural Numbers are whole, non-negative numbers, denoted by This have the right to mean either "Counting Numbers", with  = 1, 2, 3, ..., or "Whole Numbers", with  = 0, 1, 2, 3, ....

Integers are the set of totality numbers and also their opposites. These numbers deserve to be negative, positive, or zero. Integers are denoted by , with  = ..., -3, -2, -1, 0, +1, +2, +3, ....

Real Numbers are denoted through the letter . Hopeful or negative, huge or small, whole numbers or decimal numbers space all genuine Numbers. A real number is any positive or an unfavorable number. This consists of all integers and also all rational and also irrational numbers. Rational numbers, denoted by 

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, might be expressed as a fraction (such as 7/8) and irrational numbers might be to express by an boundless decimal representation (3.1415926535...). This numbers are dubbed "Real Numbers" since they are not imaginary Numbers.

An Imaginary Number is a number which once squared, offers a an adverse result. There is such a number, called i, which when squared, equals an adverse 1. This is shown below:

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When us take the square source of i, we gain this algebraic result:

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Thus, i is same to the square source of an adverse 1. Imaginary number are characterized as part of the Complex Numbers as displayed below.

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In short, a Complex Number is a number of the form a+bi where a and b are genuine numbers and i is the square source of -1.

The meanings of this numbers may be somewhat elaborate. However, the essential thing to establish is the each form of number listed above is an infinite set, and that set-builder notation is often used to explain such sets. Let"s look in ~ some instances of set-builder notation.

Example 4ReadMeaning
  K the set of all k in , such that k is any number higher than 5all integers better than 5

Note that we could additionally write this set as 6, 7, 8, .... Therefore, we deserve to say that  k > 5 = 6, 7, 8, ..., and that these sets room equal. Set-Builder Notation is also useful when working v an term of numbers, as presented in the examples below.

ExampleSet-Builder NotationReadAlso written As
5 { q | 2 q q in , together that q is any kind of number between 2 and also 63, 4, 5
6  p the collection of all p in  such that p is any type of number between 2 and also 6, inclusive.2, 3, 4, 5, 6
7  n the set of all n in  such that n is any type of number higher than or equal to 2 and less 보다 6.2, 3, 4, 5

Why usage set-builder notation?

You might be wondering about the need for such facility notation. If you have actually the set of every integers in between 2 and 6, inclusive, you might simply use roster notation to write 2, 3, 4, 5, 6, i beg your pardon is probably less complicated than utilizing set-builder notation:

 q : 2 ≤ q ≤ 6

But how would you list the Real Numbers in the same interval? making use of roster notation doesn"t make much sense in this case:

2, 2.1, 2.01, 2.001, 2.0001, ... ???

To express the collection of real numbers above, it is better to usage set-builder notation. Begin with all actual Numbers, then border them to the interval between 2 and also 6, inclusive.

 x : x ≥ 2 and x ≤ 6

You can additionally use collection builder notation come express various other sets, such as this algebraic one:

 x : x = x2 

When you evaluate this equation algebraically, friend get:

StepEvaluateExplanation
1x = x2Original equation
2x2 - x = 0Subtract x native both sides
3x(x-1) = 0Solve because that x to discover the root of this equation
4x =0 or x - 1 = 0If the product that two determinants is zero, then each aspect can be set equal to zero.
5x - 1 = 0For the 2nd factor, add 1 to both sides
5x = 0 or x = 1Solution 0, 1

Thus  x : x = x2  = 0, 1

Summary: Set-builder notation is a shorthand used to create sets, frequently for sets v an infinite number of elements. The is supplied with common types of numbers, such together integers, genuine numbers, and also natural numbers. This notation can additionally be supplied to to express sets with an expression or an equation.

Exercises

Directions: check out each concern below. Pick your prize by clicking on its button. Feedback to your answer is detailed in the outcomes BOX. If you make a mistake, rethink your answer, then pick a different button.

1.Which of the complying with sets is same to the given set below?

{ q | -4 ≤ q   -4, -3, -2, -1, 0, +1, +2, +3 -3, -2, -1, 0, 1, 2, 3 -4, -3, -2, -1, 0, +1, +2 None that the above.

RESULTS BOX: 

2.Which of the adhering to accurately describes the meaning of the given collection below?

 x : x ≥ 4

  The set of all x in  such that x is any number higher than 4 The set of all x in  such that x is any kind of number better than or equal to 4 The collection of all x in  such that x is any number higher than or equal to 4.1 None that the above.

RESULTS BOX: 

3.Which that the complying with represents the given set below?

{ n | n  The collection of all n in  such that n is any kind of number much less than 2 The collection of all n in  such that n is any type of number less than or same to 1 ..., -3, -2, -1, 0, +1 All of the above.

RESULTS BOX: 

4.Which that the complying with sets deserve to be rewritten utilizing set-builder notation? 
  apples, oranges, bananas, pears a, b, c, .., x, y, z 
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 None the the above.

RESULTS BOX: 

5.

See more: How Many Centigrams Are In A Kilogram To Centigram, Kilogram To Centigram Calculator (Kg To Cg)

Which the the adhering to is the correct set-builder notation because that the given collection below?

1.1, 2.2, 3.3, 4.4, 5.5, 6.6, 7.7, 8.8, 9.9 

   m : 1 ≤ m ≤ 9   m
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 1 ≤ m ≤ 9   m
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 1 ≤ m ≤ 9  None that the above.

RESULTS BOX: