Assumptions is an option for functions such as Simplify, Refine, and Integrate that specifies default assumptions to be made about symbolic quantities.

$Assumptions is the default setting for the Assumptions option used in such functions as Simplify, Refine, and Integrate.

Mathematica normally makes as few assumptions as possible about the objects you ask it to manipulate. This means that the results it gives are as general as possible. But ...

Simplifying with assumptions. Mathematica does not automatically simplify this, since it is only true for some values of x. Sqrt[x^2] is equal to x for x≥0, but not otherwise.

Mathematica has a flexible system for specifying arbitrary symbolic assumptions about variables. It uses a wide range of sophisticated algorithms to infer the consequences of ...

Assuming[assum, expr] evaluates expr with assum appended to $Assumptions, so that assum is included in the default assumptions used by functions such as Refine, Simplify, and ...

Refine[expr, assum] gives the form of expr that would be obtained if symbols in it were replaced by explicit numerical expressions satisfying the assumptions assum. ...

Element[x, dom] or x \[Element] dom asserts that x is an element of the domain dom. Element[{x_1, x_2, ...}, dom] asserts that all the x_i are elements of dom. Element[patt, ...

DistributionParameterAssumptions[dist] gives a logical expression for assumptions on parameters in the symbolic distribution dist.

ConditionalExpression[expr, cond] is a symbolic construct that represents the expression expr when the condition cond is True.