could it be explained please when you answer something in set builder notation and when you answer something in interval notation. I think I understand the differences between them, but don't understand why there are both of them and when they are needed. Thank you.
So, do you just want to know the difference between builder notation and interval notation and why they are needed?
Set Builder Notation Examples:
{y : y > 0} The set of all y such that y is greater than 0. Any value greater than 0.
{y : y ≠ 15} The set of all y such that y is any number except 15. Any value except 15.
{y : y < 7} The set of all y such that y is any number less than 7. Any value less than 7.
Common Set Notation:
|A|, called cardinality of A, denotes the number of elements of A. For example, if A={(1,2),(3,4)}, then |A|=2. A=B if and only if they have precisely the same elements. For example, if A={4,9} and B={n2:n=2 or n=3}, then A=B.
yes, thank you, I know HOW to answer in each of these, I just want to know WHEN to answer in either of these. Why do we need two different ways to notate.
This is what I know, I don't know why we need two different ways to notate but......
(This is what I know):
In linguistics and semiotics, a notation is a system of graphics or symbols, characters and abbreviated expressions, used (for example) in artistic and scientific disciplines to represent technical facts and quantities by convention.
This is why we use notations. (Sorry I don't know the answer to your question.)
Of course! I'd be happy to explain the differences between set builder notation and interval notation, as well as when and why they are used.
Set builder notation is a way of describing a set of numbers by providing a rule or condition that all the elements in the set must satisfy. It consists of three main components: the variable, the condition, and the variable's domain. For example, if we wanted to describe the set of all even numbers, we could use set builder notation as {x | x is an even number}. In this notation, the variable "x" represents any element in the set, the condition "x is an even number" specifies the property that the elements must have, and the vertical bar "|" separates the variable and the condition. The variable's domain is usually implied, such as "x is a natural number" or "x is a real number".
Interval notation, on the other hand, is a way of describing a set of numbers by indicating the interval or range of values that the elements can take. It consists of using brackets or parentheses to denote whether the interval includes or excludes the endpoints. There are four possible notations:
1. Open interval notation: (a, b) represents all numbers greater than "a" and less than "b" excluding both endpoints.
2. Closed interval notation: [a, b] represents all numbers greater than or equal to "a" and less than or equal to "b" including both endpoints.
3. Half-open interval notation: (a, b] represents all numbers greater than "a" and less than or equal to "b" excluding the left endpoint, but including the right endpoint.
4. Half-closed interval notation: [a, b) represents all numbers greater than or equal to "a" and less than "b" including the left endpoint, but excluding the right endpoint.
Now, as for when and why these notations are used, it depends on the context and the specific problem you are working on.
Set builder notation is typically used when you want to describe a set using a specific condition or property. This notation allows for more flexibility and precision in describing the set's elements. It is often used in mathematical proofs, defining sets in set theory, or when the condition for including an element in the set is complex or non-numeric.
Interval notation, on the other hand, is commonly used to represent sets of real numbers or intervals on a number line. It provides a concise and efficient way to represent a range of values, especially in applications such as calculus, inequalities, or representing intervals of time or distance.
Both notations have their advantages and serve different purposes. Set builder notation emphasizes the condition or property that the elements must satisfy, while interval notation focuses on the range of values the elements can take. The choice of notation depends on the context and the specific problem you are solving.
I hope this explanation helps clarify the differences and when to use them! Let me know if you have any further questions.