assumeAlso

Add permanent assumption

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

assumeAlso(condition)
assumeAlso(expr, set)

Description

assumeAlso(condition) adds the assumption that condition is true for all further calculations. It does not remove previous assumptions containing identifiers used in condition.

assumeAlso(expr, set) attaches the property set to the identifier or expression x. It does not remove previous assumptions containing identifiers used in expr.

Assumptions are mathematical conditions that are assumed to hold true for all calculations. By default, all MuPAD® identifiers are independent of each other and can take any value in the complex plane. For example, sign(1 + x^2) cannot be simplified any more because x MuPAD assumes that x is a complex number. If you set an assumption that x is a real number, then MuPAD can simplify sign(1 + x^2) to 1.

For this reason, many MuPAD functions return very general or piecewise-defined results depending on further conditions. For example, solve or int can return piecewise results.

Many mathematical theorems hold only under certain conditions. For example, x^b*y^b=(x*y)^b holds if b is an integer. But this equation is not true for all combinations of x, y, and b. For example, it is not true if x = y = -1, b = 1/2. In such cases, you can use assumptions to get more specific results.

If you use assumeAlso inside a function or procedure, MuPAD uses that assumption only inside the function or procedure. After the function or procedure call, MuPAD removes that assumption and only keeps the assumptions that were set before the function or procedure call.

If condition is a relation (for example, x < y), then MuPAD implicitly assumes that both sides of the relation are real. See Example 4.

To delete assumptions previously set on x, use unassume(x) or delete x.

When assigning a value to an identifier with assumptions, the assigned value can be inconsistent with existing assumptions. Assignments overwrite all assumptions previously set on an identifier. See Example 5.

assumeAlso accepts any condition and Boolean combinations of conditions. See Example 7.

If expr is a list, vector, or matrix, use the syntax assumeAlso(expr, set). Here, set must be specified as one of C_, R_, Q_, Z_, N_, or an expression constructed with the set operations, such as union, intersect, or minus. set also can be a function of the Type library, for example, Type::Real, Type::Integer, Type::PosInt, and so on.

Do not use the syntaxes assumeAlso(expr in set) and assumeAlso(condition) for nonscalar expr.

Examples

Example 1

Solve this equation without any assumptions on the variable x:

solve(x^5 - x, x)

Suppose, your computations deal with real numbers only. In this case, use the assume function to set the permanent assumption that x is real:

assume(x in R_)

If you solve the same equation now, you will get three real solutions:

solve(x^5 - x, x)

If you also want to get only nonzero solutions, use assumeAlso to add the corresponding assumption:

assumeAlso(x <> 0);
solve(x^5 - x, x)

MuPAD keeps both assumptions for further computations:

getprop(x)

For further computations, delete the identifier x:

delete x

Example 2

When you use assumeAlso, MuPAD does not remove existing assumptions. Instead, it combines them with new assumptions. For example, assume that n is an integer:

assume(n, Type::Integer):
getprop(n);

Add the assumption that n is positive:

assumeAlso(n, Type::Positive):
getprop(n);

For further computations, delete the identifier n:

delete n

Alternatively, set multiple assumptions in one function call using the Boolean operator and:

assume(n, Type::Integer and Type::Positive):
getprop(n);

For further computations, delete the identifier n:

delete n

Example 3

You can set separate assumptions on the real and imaginary parts of an identifier:

assume(Re(z) > 0);
assumeAlso(Im(z) < 0):
abs(Re(z));
sign(Im(z))

For further computations, delete the identifier z:

delete z

Example 4

Using assume, set the assumption x > y. Assumptions set as relations affect the properties of both identifiers.

assume(x > y)

To see the assumptions set on identifiers, use getprop:

getprop(x);
getprop(y);

To keep an existing assumption on y and add a new one, use assumeAlso. For example, add the new assumption that y is greater than 0 while keeping the assumption that y is less than x:

assumeAlso(y > 0)
is(x^2 >= y^2)

Relations, such as x > y, imply that the involved identifiers are real:

is(x, Type::Real), is(y, Type::Real)

delete x, y:

You also can add a relational assumption where one side is not an identifier, but an expression:

assumeAlso(x > 1/y)
getprop(x);
getprop(y)

For further computations, delete the identifiers x and y:

delete x, y

Example 5

_assign and := do not check if an identifier has any assumptions. The assignment operation overwrites all assumptions:

assume(a > 0):
a := -2:
a, getprop(a)

For further computations, delete the identifier a:

delete a

Example 7

Use assume to set the assumption that the identifier a is positive:

assume(a > 0)

Now, add two new assumptions using one call to assumeAlso. To combine the assumptions, use the Boolean operator and:

assumeAlso(a in Z_ and a < 5):
is(a = 0);
is(a = 1/2);
is(a = 2);
is(a = 6);

Parameters

expr

Identifier, mathematical expression, list, vector, or matrix containing identifiers.

If expr is a list, vector, or matrix, then only the syntax assumeAlso(expr, set) is valid.

set

Property representing a set of numbers or a set returned by solve. This set can be an element of Dom::Interval, Dom::ImageSet, piecewise, or one of C_, R_, Q_, Z_, N_. It also can be an expression constructed with the set operations, such as union, intersect or minus. For more examples, see Properties.

condition

Equality, inequality, element of relation, or Boolean combination (with the operators and or or).

Return Values

Void object null() of type DOM_NULL.

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