set

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.