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In mathematical logic, more complex formulas are built from atomic formulas using logical connectives and quantifiers. For example, letting denote the set of real numbers, ∀x: x ∈ ⇒ (x+1)⋅(x+1) ≥ 0 is a mathematical formula evaluating to true in the algebra of complex numbers.
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a chemical formula. The informal use of the term formula in science refers to the general construct of a relationship between given quantities. The plural of formula can be either formulas (from the most common English plural noun form ...
In mathematical logic, a ground term of a formal system is a term that does not contain any variables. Similarly, a ground formula is a formula that does not contain any variables. In first-order logic with identity with constant symbols a {\displaystyle a} and b {\displaystyle b} , the sentence Q ( a ) ∨ P ( b ) {\displaystyle Q(a)\lor P(b ...
A term denotes a mathematical object while a formula denotes a mathematical fact. In particular, terms appear as components of a formula. A first-order term is recursively constructed from constant symbols, variables, and function symbols. An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic ...
When terms and formulas are represented as strings of symbols, these rules can be used to write a formal grammar for terms and formulas. These rules are generally context-free (each production has a single symbol on the left side), except that the set of symbols may be allowed to be infinite and there may be many start symbols, for example the ...
In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given signature. [ 1 ] [ 2 ] For example, in a signature consisting of a single binary operation , the term algebra over a set X of variables is exactly the free magma generated by X .
A key property of formulas is that they can be uniquely parsed to determine the structure of the formula in terms of its propositional variables and logical connectives. When formulas are written in infix notation , as above, unique readability is ensured through an appropriate use of parentheses in the definition of formulas.
For example, in mathematics, "or" means "one, the other or both", while, in common language, it is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called "exclusive or"). Finally, many mathematical terms are common words that are used with a completely different meaning. [100]