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In computer algebra, formulas are viewed as expressions that can be evaluated as a Boolean, depending on the values that are given to the variables occurring in the expressions. For example 8 x − 5 ≥ 3 {\displaystyle 8x-5\geq 3} takes the value false if x is given a value less than 1, and the value true otherwise.
A statement such as that predicate P is satisfied by arbitrarily large values, can be expressed in more formal notation by ∀x : ∃y ≥ x : P(y). See also frequently. The statement that quantity f(x) depending on x "can be made" arbitrarily large, corresponds to ∀y : ∃x : f(x) ≥ y. arbitrary A shorthand for the universal quantifier. An ...
Algebraic notation describes the rules and conventions for writing mathematical expressions, as well as the terminology used for talking about parts of expressions. For example, the expression 3 x 2 − 2 x y + c {\displaystyle 3x^{2}-2xy+c} has the following components:
The predicate calculus goes a step further than the propositional calculus to an "analysis of the inner structure of propositions" [4] It breaks a simple sentence down into two parts (i) its subject (the object (singular or plural) of discourse) and (ii) a predicate (a verb or possibly verb-clause that asserts a quality or attribute of the object(s)).
Sentences are then built up out of atomic sentences by applying connectives and quantifiers. A set of sentences is called a theory ; thus, individual sentences may be called theorems . To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory.
The consequence of these features is that a mathematical text is generally not understandable without some prerequisite knowledge. For example, the sentence "a free module is a module that has a basis" is perfectly correct, although it appears only as a grammatically correct nonsense, when one does not know the definitions of basis, module, and free module.
Some algebraic expressions take the form of statements that relate two expressions to one another. An equation is a statement formed by comparing two expressions, saying that they are equal. This can be expressed using the equals sign ( = {\displaystyle =} ), as in 5 x 2 + 6 x = 3 y + 4 {\displaystyle 5x^{2}+6x=3y+4} .
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.