# Help with math?

I'm a 7th grader in Algebra 1 and it's hard for me. Most of my friends are in Pre-Algebra. Anyways, we went over set-builder notation in class yesterday. Although the 1 1/2 hour lesson of inequalities was boring, I really tried to listen. What is set-builder notation, and how do you use it?

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• Anonymous

In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by stating the properties that its members must satisfy. Forming sets in this manner is also known as set comprehension, set abstraction or as defining a set's intension.

Let Φ(x) be a schematic formula in which x appears free. Set builder notation has the form {x : Φ(x)} (some write {x | Φ(x)}), denoting the set of all individuals in the universe of discourse satisfying the predicate Φ(x), that is, the set whose members are every individual x such that Φ(x) is true. Set builder notation binds the variable x and must be used with the same care applied to variables bound by quantifiers.

Let {S : S is a set and S does not belong to S} denote the set of all sets that do not belong to themselves. This set cannot exist; Russell's paradox explains why. At this point, mathematics and first order logic part company.

Solutions to the paradox restrict set -builder notation in certain ways. Let X={x in A : P(x)} denote the set of every element of A satisfying the predicate P(x). The canonical restriction on set builder notation asserts that X is a set only if A is already known to be a set. This restriction is codified in the axiom schema of separation present in standard axiomatic set theory. Note that this axiom schema excludes {S : S is a set and S does not belong to S} from sethood.

The notation can be complicated, especially as in the previous example, and abbreviations are often employed when context indicates the nature of a variable. For example:

{x : x > 0}, in a context where the variable x is used only for real numbers, indicates the set of all positive real numbers;

{p/q : q is not zero}, in a context where the variables p and q are used only for integers, indicates the set of all rational numbers; and

{S : S does not belong to S}, in a context where the variable S is used only for sets, indicates the set of all sets that don't belong to themselves.

As the last example shows, such an abbreviated notation again might not denote an actual nonparadoxical set, unless there is in fact a set of all objects that might be described by the variable in question.

Another variation on set-builder notation describes the members of the set in terms of members of some other set. Specifically, {F(x) : x in A}, where F is a function symbol and A is a previously defined set, indicates the set of all values of members of A under F. For example:

{2n : n in N}, where N is the set of all natural numbers, is the set of all even natural numbers.

In axiomatic set theory, this set is guaranteed to exist by the axiom schema of replacement.

These notations can be combined in the form {F(x) : x in A, P(x)}, which indicates the set of all values under F of those members of A that satisfy P. For example:

{p/q : p in Z, q in Z, q is not zero}, where Z is the set of all integers, is the set of all rational numbers (Q).

This example also shows how multiple variables can be used (both p and q in this case). This notation is acceptable even though e.g. 2/3 and 4/6 are both included in this definition, and a set can not contain multiple copies of an element; the case p=4, q=6 says with harmless redundancy that 2/3 is in the set.

Set-builder notation is closely related to a construct in some programming languages, most notably Python and Haskell, called list comprehension.

In Python, list comprehensions are denoted by square brackets, and have a different syntax to set-builder, but are fundamentally the same. Consider these examples, given in both set-builder notation and Python list comprehension.

Set-builder:

{l: l in L}

{{k, x}: k in K and x in X if P(x)

List comprehension:

[l for l in L]

[(k, x) for k in K for x in X if P(x)]

Note: While Python's list comprehension works similarly to set-builder notation, it does not denote a set but rather creates a mathematical tuple (as opposed to Python's native tuple datatype; the actual returned value's type is list) based on existing tuples. It is possible to use true sets in Python with the set keyword and set class, but this causes additional deviations from set-builder notation.

• 1 decade ago

Set builder notation is just a way of writing down the information that is needed to describe and build a set. It is really useful for large logic problems especially when building software that is going to use sets. It will be used in computer science.

Take a look at this....

http://en.wikipedia.org/wiki/Set-builder_notation

• 1 decade ago

Basically is a way to write the range that the graph is in. So lets say that the graph starts at (-5, 2) and it ends at (7,9) the you can write for the x values [-5,7] (always from smallest to biggest) or {x l -5< x < 7} both are correct. and same thing for the y values. And its commomly known as set notations.

• corl
Lv 4
4 years ago

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• 1 decade ago

A shorthand used to write sets, often sets with an infinite number of elements.

Note: The set {x : x > 0} is read aloud, "the set of all x such that x is greater than 0." It is read aloud exactly the same way when the colon : is replaced by the vertical line | as in {x | x > 0}.

• 1 decade ago

Shoot, I've finished 4 levels of college calculus and don't know what set-builder notation is.

• 1 decade ago

are you SERIOUS!?

Im in algebra 1 and im in the 9th grade...lol

Ok, so, anything that has to do with math, or school is gonna be boring. Just think about it this way, scince your a year ahead of your friends, you get to get out of the math a year early!

Lol

<3 kat