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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]
An algebraic equation is an equation involving polynomials, for which algebraic expressions may be solutions. If you restrict your set of constants to be numbers, any algebraic expression can be called an arithmetic expression. However, algebraic expressions can be used on more abstract objects such as in Abstract algebra.
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.
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of statements within those systems.It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations such as addition and multiplication.
One stark difference is that Diophantine equations have an undecidable solubility problem, [2] whereas the analogous problem for word equations is decidable. [ 3 ] A classical example of a word equation is the commutation equation x w = ⋅ w x {\displaystyle xw{\overset {\cdot }{=}}wx} , in which x {\displaystyle x} is an unknown and w ...
For example, the word "encyclopedia" is a sequence of symbols in the English alphabet, a finite set of twenty-six letters. Since a word can be described as a sequence, other basic mathematical descriptions can be applied. The alphabet is a set, so as one would expect, the empty set is a subset. In other words, there exists a unique word of ...
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