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assume

Set permanent assumption

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

assume(condition)
assume(expr, set)

Description

assume(condition) sets the assumption that condition condition is true for all further calculations. This call removes all previous assumptions containing identifiers used in condition.

assume(expr, set) attaches the property set to the identifier or expression expr. This call overwrites all 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 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 that depend 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 assume inside a function or procedure, MuPAD uses the new assumption and ignores existing assumptions only inside the function or procedure. After the function or procedure call, MuPAD removes the new assumption and restores 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 you assign 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.

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

If expr is a list, vector, or matrix, use the syntax assume(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 assume(expr in set) and assume(condition) for nonscalar expr.

Examples

Example 1

Set an assumption that identifier n is an integer. Here, assume(n, Type::Integer) is equivalent to assume(n in Z_) because n is a scalar.

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

Check if n^2 is a nonnegative integer. MuPAD uses the assumption that you set on n.

is(n^2, Type::NonNegInt)

Other system functions take this assumption into account:

abs(n^2 + 1);
simplify(sin(2*n*PI))

For further computations, delete the identifier n:

delete n

Example 2

To keep the existing assumptions and combine them with the new ones, use assumeAlso:

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

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

For further computations, delete the identifier n:

delete n

Alternatively, set multiple assumptions in one function call:

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))

is(z, Type::Real), is(z > 0)

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)

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

assume(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 6

Set the assumption that x is positive and find the absolute value of x, the sign of x, and the real and imaginary parts of x. These system functions take assumptions set on identifiers into account:

assume(x > 0):
abs(x), sign(x), Re(x), Im(x)

Try expanding the expression ln(z1*z2) without additional assumptions. It does not expand because ln(z1*z2) = ln(z1) + ln(z2) is not true for arbitrary z1, z2 in the complex plane:

expand(ln(z1*z2))

Now, set the assumption that one number is real and positive. Expand the same expression:

assume(z1 > 0): expand(ln(z1*z2))

For further computations, remove the assumptions on x and z1:

unassume(x); unassume(z1)

Example 7

Set these two assumptions on the identifier a. To combine the assumptions, use the Boolean operator and:

assume(a > 0 and a in Z_):
is(a = 0);
is(a = 1/2);
is(a = 2);

Parameters

expr

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

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

set

Property representing a set of numbers or a set returned by solve.

For example, 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.

See Also

MuPAD Functions

Related Examples

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