What Is the Meaning of Unbounded & Bounded in Math?

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There are very few people who possess the innate ability to figure out math problems with ease. The rest sometimes need help. Mathematics has a large vocabulary which can becoming confusing as more and more words are added to your lexicon, especially because words can have different meanings depending on the branch of math being studied. An example of this confusion exists in the word pair "bounded" and "unbounded."

Functions

The primary usage of the words "bounded" and "unbounded" in mathematics occurs in the terms "bounded function" and "unbounded function." A bounded function is one that can be contained by straight lines along the x-axis in a graph of the function. For example, sine waves are functions that are considered bounded. One that does not have a maximum or minimum x-value, is called unbounded. In terms of mathematical definition, a function "f" defined on a set "X" with real/complex values is bounded if its set of values is bounded.

Operators

In functional analysis, there is another usage for the terms "bounded" and "unbounded." You can have bounded and unbounded operators. These operators are different and often not compatible with the definition of bounded for functions. From Springer Online Reference Works' Encyclopaedia of Mathematics, an unbounded operator is "a mapping A from a set M in a topological vector space X into a topological vector space Y such that there is a bounded set N ⊂ M whose image A(N) is an unbounded set in Y."

Sets

You can also have a bounded and unbounded set of numbers. This definition is much simpler, but remains similar in meaning to the previous two. A bounded set is a set of numbers that has an upper and a lower bound. For example, the interval [2,401) is a bounded set, because it has a finite value at both ends. Also, you could have a bounded set of numbers like this: {1,1/2,1/3,1/4...}, An unbounded set would have the opposite characteristics; its upper and/or lower bounds would not be finite.

Meaning

In the above three most common ways of using the terms "bounded" and "unbounded" in mathematics, there are some common characteristics that can be used if you come across the term in an unfamiliar setting. Generally, and by definition, things that are bounded can not be infinite. A bounded anything has to be able to be contained along some parameters. Unbounded means the opposite, that it cannot be contained without having a maximum or minimum of infinity.

References

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