Term (logic)

This article explains the mathematical term. For other meanings, see Term (disambiguation).

In mathematics, a term is a meaningful combination of numbers, variables, symbols for mathematical operations and parentheses. The starting point is the atomic terms, to which all numbers (constants) and variables belong. Terms can be seen as the syntactically correctly formed words or groups of words in the formal language of mathematics.

In practice, the term is often used to talk about individual components of a formula or larger term. For example, for the linear function , one can f(x)=mx+b talk about a linear term and a constant term

Colloquial explanation

The term "term" is used colloquially for everything that carries a meaning. In the narrower sense, it refers to mathematical entities that can in principle be calculated, at least if you have assigned values to the variables they contain. For example, is $(x+y)^{2}$ a term, because if you assign a value to the variables and contained in it, the term also gets a value. Instead of numbers, other mathematical objects may be considered here, for example is $(p_{1}\vee \neg p_{2})\wedge p_{3}$ a term that obtains a value if one assigns a truth value to the Boolean variables . $p_{1},p_{2},p_{3}$ However, in the normal case (one-sort logic), the exact mathematical definition makes no reference to the possible value assignments, as explained below.

Roughly, we can say that a term is a side of an equation or relation, such as an inequality. The equation or relation itself is not a term, it consists of terms.

The following operations can usually be performed with terms:

(to do this, first calculate the "inner" functions and then the outer): $(2+3)^{2}=5^{2}=25$
transform according to certain calculation rules: $(x+y)^{2}=x^{2}+2xy+y^{2}$ by applying the distributive law and some other "allowed" rules.
with each other if relations are defined for the matching types: $2xy\leq x^{2}+y^{2}$
into each other (often a term is substituted for a variable of another term). A special form of substitution is substitution, where a term with variables is replaced by another term with variables (usually a single variable): $(x+y)^{2}$ arises from $z^{2}$ by substituting for .

Terms or subterms are often named according to their substantive meaning. In the term ${\tfrac {1}{2}}mv^{2}+mgh$ , which in physics describes the total energy of a mass point, the first summand is called the "kinetic energy term" and the second the "potential energy term". Characteristic properties are also often used for naming. For example, the "quadratic term" in $x^{3}+7x^{2}-2x+1$ means the subterm , $7x^{2}$ because this is the subterm containing the variable in squared form.

Applications

If one forms a term with variables, then one often intends in applications a replacement of these variables by certain values, which originate from a certain basic set or definition set. For the notion of the term itself, the specification of such a set according to the above formal definition is not necessary. Then one is no longer interested in the abstract term, but in a function defined by this term in a certain model.

Thus, a rule of thumb for calculating the stopping distance (braking distance plus reaction distance) of a car in meters $\left({\tfrac {x}{10}}\right)^{2}+\left({\tfrac {x}{10}}\cdot 3\right)$ . This string is a term. We intend to substitute for the speed of the car in km per hour, to use the value that the term then takes as the braking distance in meters. For example, if a car is traveling 160 km/h, the formula yields $\left({\tfrac {160}{10}}\right)^{2}+\left({\tfrac {160}{10}}\cdot 3\right)$ a stopping distance of 304 m.

We use the term here to define the assignment rule of a function $f\colon \mathbb{R} _{0}^{+}\to \mathbb{R} _{0}^{+}$ , $x\mapsto \left({\tfrac {x}{10}}\right)^{2}+\left({\tfrac {x}{10}}\cdot 3\right)$ .

Terms themselves are neither true nor false and have no values. Only in a model, i.e. with specification of a basic set for the occurring variables, terms can assume values.

Term (logic)

Colloquial explanation

Applications

Search by letter