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3.3.10 Using Assumptions
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 sometimes these results are considerably more complicated than they would be if more assumptions were made.
Doing operations with assumptions.
Simplify by default does essentially nothing with this expression.
In[1]:= Simplify[1/Sqrt[x] - Sqrt[1/x]]
Out[1]= 
The reason is that its value is quite different for different choices of  .
In[2]:= % /. x -> {-3, -2, -1, 1, 2, 3}
Out[2]= 
With the assumption  , Simplify can immediately reduce the expression to 0.
In[3]:= Simplify[1/Sqrt[x] - Sqrt[1/x], x > 0]
Out[3]= 
Without making assumptions about  and  , nothing can be done.
In[4]:= FunctionExpand[Log[x y]]
Out[4]= 
If  and  are both assumed positive, the log can be expanded.
In[5]:= FunctionExpand[Log[x y], x > 0 && y > 0]
Out[5]= 
By applying Simplify and FullSimplify with appropriate assumptions to equations and inequalities you can in effect establish a vast range of theorems.
Without making assumptions about  the truth or falsity of this equation cannot be determined.
In[6]:= Simplify[Abs[x] == x]
Out[6]= 
Now Simplify can prove that the equation is true.
In[7]:= Simplify[Abs[x] == x, x > 0]
Out[7]= 
This establishes the standard result that the arithmetic mean is larger than the geometric one.
In[8]:= Simplify[(x + y)/2 >= Sqrt[x y], x >= 0 && y >= 0]
Out[8]= 
This proves that  lies in the range  for all positive arguments.
In[9]:= FullSimplify[0 < Erf[x] < 1, x > 0]
Out[9]= 
An important class of assumptions are those which assert that some object is an element of a particular domain. You can set up such assumptions using x  dom, where the  character can be entered as  elem or \[Element].
Asserting that objects are elements of domains.
This confirms that  is an element of the domain of real numbers.
In[10]:= Pi  Reals
Out[10]= 
These numbers are all elements of the domain of algebraic numbers.
In[11]:= {1, Sqrt[2], 3 + Sqrt[5]} Algebraics
Out[11]= 
Mathematica knows that  is not an algebraic number.
In[12]:= Pi Algebraics
Out[12]= 
Current mathematics has not established whether  is an algebraic number or not.
In[13]:= E + Pi Algebraics
Out[13]= 
This represents the assertion that the symbol x is an element of the domain of real numbers.
In[14]:= x Reals
Out[14]= 
Domains supported by Mathematica.
If  is assumed to be an integer,  is zero.
In[15]:= Simplify[Sin[n Pi], n Integers]
Out[15]= 
This establishes the theorem  if  is assumed to be a real number.
In[16]:= Simplify[Cosh[x] >= 1, x Reals]
Out[16]= 
If you say that a variable satisfies an inequality, Mathematica will automatically assume that it is real.
In[17]:= Simplify[x Reals, x > 0]
Out[17]= 
By using Simplify, FullSimplify and FunctionExpand with assumptions you can access many of Mathematica's vast collection of mathematical facts.
This uses the periodicity of the tangent function.
In[18]:= Simplify[Tan[x + Pi k], k Integers]
Out[18]= 
The assumption k/2  Integers implies that k must be even.
In[19]:= Simplify[Tan[x + Pi k/2], k/2 Integers]
Out[19]= 
Mathematica knows that  for positive  .
In[20]:= Simplify[Log[x] < Exp[x], x > 0]
Out[20]= 
FullSimplify accesses knowledge about special functions.
In[21]:= FullSimplify[Im[BesselJ[0, x]], x Reals]
Out[21]= 
Mathematica knows about discrete mathematics and number theory as well as continuous mathematics.
This uses Wilson's Theorem to simplify the result.
In[22]:= FunctionExpand[Mod[(p - 1)!, p], p Primes]
Out[22]= 
This uses the multiplicative property of the Euler phi function.
In[23]:= FunctionExpand[EulerPhi[m n], {m, n} Integers &&
GCD[m, n] == 1]
Out[23]= 
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