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The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word problems for groups and semigroups can be phrased as word problems for algebras. [1]
In mathematics, a variable (from Latin variabilis, "changeable") is a symbol, typically a letter, that refers to an unspecified mathematical object. [1] [2] [3] One says colloquially that the variable represents or denotes the object, and that any valid candidate for the object is the value of the variable.
Word problem from the Līlāvatī (12th century), with its English translation and solution. In science education, a word problem is a mathematical exercise (such as in a textbook, worksheet, or exam) where significant background information on the problem is presented in ordinary language rather than in mathematical notation.
Word problem (mathematics education), a type of textbook exercise or exam question to have students apply abstract mathematical concepts to real-world situations; Word problem (mathematics), a decision problem for algebraic identities in mathematics and computer science; Word problem for groups, the problem of recognizing the identity element ...
The word problem was one of the first examples of an unsolvable problem to be found not in mathematical logic or the theory of algorithms, but in one of the central branches of classical mathematics, algebra. As a result of its unsolvability, several other problems in combinatorial group theory have been shown to be unsolvable as well.
In group theory, a word is any written product of group elements and their inverses. For example, if x , y and z are elements of a group G , then xy , z −1 xzz and y −1 zxx −1 yz −1 are words in the set { x , y , z }.
A random deviate or simply deviate is the difference of a random variate with respect to the distribution central location (e.g., mean), often divided by the standard deviation of the distribution (i.e., as a standard score). [1] Random variates are used when simulating processes driven by random influences (stochastic processes).
Many word problems are undecidable based on the Post correspondence problem. Any two homomorphisms, with a common domain and a common codomain form an instance of the Post correspondence problem, which asks whether there exists a word in the domain such that () = (). Post proved that this problem is undecidable; consequently, any word problem ...