set

(noun)

A collection of zero or more objects, possibly infinite in size, and disregarding any order or repetition of the objects that may be contained within it.

Related Terms

  • subset
  • superset
  • finite
  • natural numbers
  • sequence
  • expression

(noun)

A collection of unique objects, potentially infinite in size, that is not reliant on the order of the objects contained within it.

Related Terms

  • subset
  • superset
  • finite
  • natural numbers
  • sequence
  • expression

Examples of set in the following topics:

  • Sets of Numbers

    • Sets are one of the most fundamental concepts in mathematics.
    • The second way of describing a set is through extension: listing each member of the set.
    • All set operations preserve this property.
    • A subset is a set whose every element is also contained in another set.
    • For example, if every member of set $A$ is also a member of set $B$, then $A$ is said to be a subset of $B$.
  • Interval Notation

    • Interval notation uses parentheses and brackets to describe sets of real numbers and their endpoints.
    • A "real interval" is a set of real numbers such that any number that lies between two numbers in the set is also included in the set.
    • Other examples of intervals include the set of all real numbers and the set of all negative real numbers.
    • The set of all real numbers is the only interval that is unbounded at both ends; the empty set (the set containing no elements) is bounded.
    • Use interval notation to show how a set of numbers is bounded
  • Averages

    • The arithmetic mean, or average, of a set of numbers indicates the "middle" or "typical" value of a data set.
    • This is simply a mathematical way of writing "the mean equals the sum of all of the values in the set, divided by the number of values in the set."
    • To see how this applies to an actual set of numbers, consider the following set: $\{3,5,10\}$.
    • Next, divide their sum by 3, the number of values in the set:
    • Therefore, the average of the set of numbers $\{3,5,10\}$ is 5.
  • Solving Systems Graphically

    • Some systems have only one set of correct answers, while others have multiple sets that will satisfy all equations.
    • Shown graphically, a set of equations solved with only one set of answers will have only have one point of intersection, as shown below.
    • In a set of linear equations (such as in the image below), there is only one solution.
    • To determine the solutions of the set of equations, identify the points of intersection between the graphed equations.
    • This graph shows a system of equations with two variables and only one set of answers that satisfies both equations.
  • Permutations of Nondistinguishable Objects

    • Recall that the number of possible permutations of a set of $n$ distinct elements is given by $n!
    • This rule holds true for sets of any size, so long as the elements are all distinct.
    • For example, given the numbers $1$, $3$, and $3$ in a set, there are 2 ways to obtain the order $(3,1,3)$.
    • Thus, the number of possible distinct permutations in the set is:
    • Calculate the number of permutations of a given set of objects, some being nondistinguishable
  • Permutations of Distinguishable Objects

    • The number of permutations of distinct elements can be calculated when not all elements from a given set are used.
    • Recall that, if all objects in a set are distinct, then they can be arranged in $n!
    • However, consider a situation where not all of the elements in a set of distinct objects are used in each permutation.
    • If not all of the objects in a set of unique elements are chosen, the following formula is used.
    • This formula determines the number of possible permutations of $k$ elements selected from the set of $n$ elements:
  • Total Number of Subsets

    • The total number of subsets is the number of sets with 0 elements, 1 element, 2 elements, etc.
    • These numbers also arise in combinatorics, where $n^b$ gives the number of different combinations of $b$ elements that can be chosen from an $n$-element set.
    • The total number of subsets is the number of sets with 0 elements, 1 element, 2 elements, etc.
    • The total number of subsets of a set with $n$ elements is $2^n$.
    • For example, how many subsets are in the set: $\left \{ P, Q, R, S, T, U \right \}$ ?
  • Introduction to Domain and Range

    • The domain of a function is the set of all possible input values that produce some output value range
    • The domain of a function is the set of input values, $x$, for which a function is defined.
    • It is important to note that not all functions have the set of real numbers as their domain.
    • In this case, the domain of $f$ is the set of all real numbers except $0$.
    • So this function's domain is the set of all real numbers such that $x>1$ and $x\neq5$.
  • Functions and Their Notation

    • A function maps a set of inputs onto a set of permissible outputs.
    • In mathematics, a function is a relation between a set of inputs and a set of permissible outputs.
    • Another commonly used notation for a function is $f:X\rightarrow Y$, which reads as saying that $f$ is a function that maps values from the set $X$ onto values of the set $Y$.
    • A relation is a connection between values in one set and values in another.
  • Zeroes of Polynomial Functions With Rational Coefficients

    • The set of all rational numbers is usually denoted by a boldface Q (or Unicode ℚ).
    • Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.
    • The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers.
    • By setting each term to zero, it can be found that the zeros for this equation are x=-6 and x=-9/2.
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