Sage Introductory Programming Tutorial

This Sage document is one of the tutorials developed for the MAA PREP Workshop “Sage: Using Open-Source Mathematics Software with Undergraduates” (funding provided by NSF DUE 0817071). It is licensed under the Creative Commons Attribution-ShareAlike 3.0 license (CC BY-SA).

This tutorial will cover the following topics (and various others throughout):

We will motivate our examples using basic matrices and situations one might want to handle in everyday use of matrices in the classroom.

Methods and Dot Notation

Making a new matrix is not too hard in Sage.

sage: A = matrix([[1,2],[3,4]])

>>> from sage.all import *
>>> A = matrix([[Integer(1),Integer(2)],[Integer(3),Integer(4)]])

A = matrix([[1,2],[3,4]])

As we can see, this gives the matrix with rows given in the two bracketed sets.

sage: A
[1 2]
[3 4]

>>> from sage.all import *
>>> A
[1 2]
[3 4]

A

Some commands are available right off the bat, like derivatives are. This is the determinant.

sage: det(A)
-2

>>> from sage.all import *
>>> det(A)
-2

det(A)

But some things are not available this way - for instance a row-reduced echelon form. We can ‘tab’ after this to make sure.

sage: r

>>> from sage.all import *
>>> r

r

So, as we’ve already seen in previous tutorials, many of the commands in Sage are “methods” of objects.

That is, we access them by typing:

  • the name of the mathematical object,

  • a dot/period,

  • the name of the method, and

  • parentheses (possibly with an argument).

This is a huge advantage, once you get familiar with it, because it allows you to do only the things that are possible, and all such things.

First, let’s do the determinant again.

sage: A.det()
-2

>>> from sage.all import *
>>> A.det()
-2

A.det()

Then we do the row-reduced echelon form.

sage: A.rref()
[1 0]
[0 1]

>>> from sage.all import *
>>> A.rref()
[1 0]
[0 1]

A.rref()

It is very important to keep in the parentheses.

Note

Things that would be legal without them would be called ‘attributes’, but Sage prefers stylistically to hide them, since math is made of functions and not elements of sets. Or so a category-theorist would say.

sage: A.det # Won't work
<built-in method det of sage.matrix.matrix_integer_dense.Matrix_integer_dense object at ...>

>>> from sage.all import *
>>> A.det # Won't work
<built-in method det of sage.matrix.matrix_integer_dense.Matrix_integer_dense object at ...>

A.det # Won't work

This is so useful because we can use the ‘tab’ key, remember!

sage: A.

>>> from sage.all import *
>>> A.

A.

Sometimes you will have surprises. Subtle changes in an object can affect what commands are available, or what their outcomes are.

sage: A.echelon_form()
[1 0]
[0 2]

>>> from sage.all import *
>>> A.echelon_form()
[1 0]
[0 2]

A.echelon_form()

This is because our original matrix had only integer coefficients, and you can’t make the last entry one via elementary operations unless you multiply by a rational number!

sage: B = A.change_ring(QQ); B.echelon_form()
[1 0]
[0 1]

>>> from sage.all import *
>>> B = A.change_ring(QQ); B.echelon_form()
[1 0]
[0 1]

B = A.change_ring(QQ); B.echelon_form()

Another question is whether one needs an argument. Remember, it’s easy to just read the documentation!

Below, let’s see whether we need an argument to get a column.

sage: A.column?

>>> from sage.all import *
>>> A.column?

A.column?

It looks like we do. Let’s input 1.

sage: A.column(1)
(2, 4)

>>> from sage.all import *
>>> A.column(Integer(1))
(2, 4)

A.column(1)

Notice that this gives the SECOND column!

sage: A
[1 2]
[3 4]

>>> from sage.all import *
>>> A
[1 2]
[3 4]

A

What is that about?

Lists, Loops, and Set Builders

(Especially List Comprehensions!)

In the previous example, we saw that the 1 choice for the column of a matrix gives the second column.

sage: matrix([[1,2],[3,4]]).column(1)
(2, 4)

>>> from sage.all import *
>>> matrix([[Integer(1),Integer(2)],[Integer(3),Integer(4)]]).column(Integer(1))
(2, 4)

matrix([[1,2],[3,4]]).column(1)

You might have thought that the would give the first column, but Sage (along with the Python programming language) begins numbering of anything that is like a sequence at zero. We’ve mentioned this once before, but it’s very important to remember.

To reinforce this, let’s formally introduce a fundamental object we’ve seen once or twice before, called a list.

You should think of a list as an ordered set, where the elements of the set can be pretty much anything - including other lists.

sage: my_list=[2,'Grover',[3,2,1]]; my_list
[2, 'Grover', [3, 2, 1]]

>>> from sage.all import *
>>> my_list=[Integer(2),'Grover',[Integer(3),Integer(2),Integer(1)]]; my_list
[2, 'Grover', [3, 2, 1]]

my_list=[2,'Grover',[3,2,1]]; my_list

You can access any elements of such a list quite easily using square brackets. Just remember that the counting starts at zero.

sage: my_list[0]; my_list[2]
2
[3, 2, 1]

>>> from sage.all import *
>>> my_list[Integer(0)]; my_list[Integer(2)]
2
[3, 2, 1]

my_list[0]; my_list[2]

There are lots of advanced things one can do with lists.

sage: my_list[0:2]
[2, 'Grover']

>>> from sage.all import *
>>> my_list[Integer(0):Integer(2)]
[2, 'Grover']

my_list[0:2]

However, our main reason for introducing this is more practical, as we’ll now see.

  • One of the best uses of the computer in the classroom is to quickly show tedious things.

  • One of the most tedious things to do by hand in linear algebra is taking powers of matrices.

  • Here we make the first four powers of our matrix ‘by hand’.

sage: A = matrix([[1,2],[3,4]])
sage: A^0; A^1; A^2; A^3; A^4
[1 0]
[0 1]
[1 2]
[3 4]
[ 7 10]
[15 22]
[ 37  54]
[ 81 118]
[199 290]
[435 634]

>>> from sage.all import *
>>> A = matrix([[Integer(1),Integer(2)],[Integer(3),Integer(4)]])
>>> A**Integer(0); A**Integer(1); A**Integer(2); A**Integer(3); A**Integer(4)
[1 0]
[0 1]
[1 2]
[3 4]
[ 7 10]
[15 22]
[ 37  54]
[ 81 118]
[199 290]
[435 634]

A = matrix([[1,2],[3,4]])
A^0; A^1; A^2; A^3; A^4

This is not terrible, but it’s not exactly nice either, particularly if you might want to do something with these new matrices.

Instead, we can do what is known as a loop construction. See the notation below; it’s at least vaguely mathematical.

sage: for i in [0,1,2,3,4]:
....:     A^i
[1 0]
[0 1]
[1 2]
[3 4]
[ 7 10]
[15 22]
[ 37  54]
[ 81 118]
[199 290]
[435 634]

>>> from sage.all import *
>>> for i in [Integer(0),Integer(1),Integer(2),Integer(3),Integer(4)]:
...     A**i
[1 0]
[0 1]
[1 2]
[3 4]
[ 7 10]
[15 22]
[ 37  54]
[ 81 118]
[199 290]
[435 634]

for i in [0,1,2,3,4]:
    A^i

What did we do?

  • For each \(i\) in the set \(\{0,1,2,3,4\}\), return \(A^i\).

Yeah, that makes sense. The square brackets created a list, and the powers of the original matrix come in the same order as the list.

(The colon in the first line and the indentation in the second line are extremely important; they are the basic syntactical structure of Python.)

For the curious: this is better, but still not perfect. It would be best to find a quicker way to write the possible values for \(i\). There are two ways to do this in Sage.

sage: for i in [0..4]:
....:     det(A^i)
1
-2
4
-8
16

>>> from sage.all import *
>>> for i in (ellipsis_range(Integer(0),Ellipsis,Integer(4))):
...     det(A**i)
1
-2
4
-8
16

for i in [0..4]:
    det(A^i)

sage: for i in range(5):
....:     det(A^i)
1
-2
4
-8
16

>>> from sage.all import *
>>> for i in range(Integer(5)):
...     det(A**i)
1
-2
4
-8
16

for i in range(5):
    det(A^i)

These ways of constructing lists are very useful - and demonstrate that, like many Sage/Python things, that counting begins at zero and ends at one less than the “end” in things like range.

Below, we show that one can get step sizes other than one as well.

sage: list(range(3, 23, 2))
[3, 5, 7, 9, 11, 13, 15, 17, 19, 21]
sage: [3,5..21]
[3, 5, 7, 9, 11, 13, 15, 17, 19, 21]

>>> from sage.all import *
>>> list(range(Integer(3), Integer(23), Integer(2)))
[3, 5, 7, 9, 11, 13, 15, 17, 19, 21]
>>> (ellipsis_range(Integer(3),Integer(5),Ellipsis,Integer(21)))
[3, 5, 7, 9, 11, 13, 15, 17, 19, 21]

list(range(3, 23, 2))
[3,5..21]

Note

It is also important to emphasize that the range command does not include its last value! For a quick quiz, confirm this in the examples above.

This all works well. However, after a short time this will seem tedious as well (you may have to trust us on this). It turns out that there is a very powerful way to create such lists in a way that very strongly resembles the so-called set builder notation, called a list comprehension .

We start with a relatively easy example:

\[\{n^2\mid n\in\ZZ, 3 \leq n \leq 12\}\]

Who hasn’t written something like this at some point in a course?

This is a natural for the list comprehension, and can be very powerful when used in Sage.

sage: [n^2 for n in [3..12]]
[9, 16, 25, 36, 49, 64, 81, 100, 121, 144]

>>> from sage.all import *
>>> [n**Integer(2) for n in (ellipsis_range(Integer(3),Ellipsis,Integer(12)))]
[9, 16, 25, 36, 49, 64, 81, 100, 121, 144]

[n^2 for n in [3..12]]

That’s it. This sort of turns the loop around.

  • The notation is easiest if you think of it mathematically; “The set of \(n^2\), for (all) \(n\) in the range between 3 and 13.”

This is phenomenally useful. Here is a nice plotting example:

sage: plot([x^n for n in [2..6]],(x,0,1))
Graphics object consisting of 5 graphics primitives

>>> from sage.all import *
>>> plot([x**n for n in (ellipsis_range(Integer(2),Ellipsis,Integer(6)))],(x,Integer(0),Integer(1)))
Graphics object consisting of 5 graphics primitives

plot([x^n for n in [2..6]],(x,0,1))

Now we apply it to the example we were doing in the first place. Notice we now have a nice concise description of all determinants of these matrices, without the syntax of colon and indentation:

sage: [det(A^i) for i in [0..4]]
[1, -2, 4, -8, 16]

>>> from sage.all import *
>>> [det(A**i) for i in (ellipsis_range(Integer(0),Ellipsis,Integer(4)))]
[1, -2, 4, -8, 16]

[det(A^i) for i in [0..4]]

Tables

Finally, getting away from strictly programming, here is a useful tip.

Some of you may be familiar with a way to take such data and put it in tabular form from other programs. The table command does this for us:

sage: table( [ (i,det(A^i)) for i in [0..4] ] )
  0   1
  1   -2
  2   4
  3   -8
  4   16

>>> from sage.all import *
>>> table( [ (i,det(A**i)) for i in (ellipsis_range(Integer(0),Ellipsis,Integer(4))) ] )
  0   1
  1   -2
  2   4
  3   -8
  4   16

table( [ (i,det(A^i)) for i in [0..4] ] )

Notice that each element of this list is two items in parentheses (a so-called tuple).

Even better, we can put a header line on it to make it really clear what we are doing, by adding lists. We’ve seen keywords like header=True when doing some of our plotting and limits. What do you think will happen if you put dollar signs around the labels in the header?

sage: table( [('i', 'det(A^i)')] + [ (i,det(A^i)) for i in [0..4] ], header_row=True)
    i   det(A^i)
  ├───┼──────────┤
    0   1
    1   -2
    2   4
    3   -8
    4   16

>>> from sage.all import *
>>> table( [('i', 'det(A^i)')] + [ (i,det(A**i)) for i in (ellipsis_range(Integer(0),Ellipsis,Integer(4))) ], header_row=True)
    i   det(A^i)
  ├───┼──────────┤
    0   1
    1   -2
    2   4
    3   -8
    4   16

table( [('i', 'det(A^i)')] + [ (i,det(A^i)) for i in [0..4] ], header_row=True)

Defining Functions

Or, Extending Sage

It is often the case that Sage can do something, but doesn’t have a simple command for it. For instance, you might want to take a matrix and output the square of that matrix minus the original matrix.

sage: A = matrix([[1,2],[3,4]])
sage: A^2-A
[ 6  8]
[12 18]

>>> from sage.all import *
>>> A = matrix([[Integer(1),Integer(2)],[Integer(3),Integer(4)]])
>>> A**Integer(2)-A
[ 6  8]
[12 18]

A = matrix([[1,2],[3,4]])
A^2-A

How might one do this for other matrices? Of course, you could just always do \(A^2-A\) again and again. But this would be tedious and hard to follow, as with so many things that motivate a little programming. Here is how we solve this problem.

sage: def square_and_subtract(mymatrix):
....:     return mymatrix^2-mymatrix

>>> from sage.all import *
>>> def square_and_subtract(mymatrix):
...     return mymatrix**Integer(2)-mymatrix

def square_and_subtract(mymatrix):
    return mymatrix^2-mymatrix

The def command has created a new function called square_and_subtract. It should even be available using tab-completion.

Here are things to note about its construction:

  • The input is inside the parentheses.

  • The indentation and colon are crucial, as above.

  • There will usually be a return value, given by return. This is what Sage will give below the input cell.

sage: square_and_subtract(A)
[ 6  8]
[12 18]

>>> from sage.all import *
>>> square_and_subtract(A)
[ 6  8]
[12 18]

square_and_subtract(A)

sage: square_and_subtract(matrix([[1.5,0],[0,2]]))
[0.750000000000000 0.000000000000000]
[0.000000000000000  2.00000000000000]

>>> from sage.all import *
>>> square_and_subtract(matrix([[RealNumber('1.5'),Integer(0)],[Integer(0),Integer(2)]]))
[0.750000000000000 0.000000000000000]
[0.000000000000000  2.00000000000000]

square_and_subtract(matrix([[1.5,0],[0,2]]))

We can get a documentation string available by putting it in triple quotes """.

sage: def square_and_subtract(mymatrix):
....:     """
....:     Return `A^2-A`
....:     """
....:     return mymatrix^2-mymatrix

>>> from sage.all import *
>>> def square_and_subtract(mymatrix):
...     """
...     Return `A^2-A`
...     """
...     return mymatrix**Integer(2)-mymatrix

def square_and_subtract(mymatrix):
    """
    Return `A^2-A`
    """
    return mymatrix^2-mymatrix
sage: square_and_subtract?

>>> from sage.all import *
>>> square_and_subtract?

square_and_subtract?

Pretty cool! And potentially quite helpful to students - and you - especially if the function is complicated. The \(A\) typesets properly because we put it in backticks (see above).

A very careful reader may have noticed that there is nothing that requires the input mymatrix to be a matrix. Sage will just try to square whatever you give it and subtract the original thing.

sage: square_and_subtract(sqrt(5))
-sqrt(5) + 5

>>> from sage.all import *
>>> square_and_subtract(sqrt(Integer(5)))
-sqrt(5) + 5

square_and_subtract(sqrt(5))

This is a typical thing to watch out for; just because you define something doesn’t mean it’s useful (though in this case it was).

Try to define a function which inputs a matrix and returns the determinant of the cube of the matrix. (There are a few ways to do this, of course!)

Gotchas from names and copies

Or, What’s in a Name

Before we finish the tutorial, we want to point out a few programming-related things that often trip people up.

The first ‘gotcha’ is that it’s possible to clobber constants!

sage: i
4

>>> from sage.all import *
>>> i
4

i

Can you figure out why i=4? Look carefully above to see when this happened.

  • This gives a valuable lesson; any time you use a name there is potential for renaming.

This may seem quite bad, but could be quite logical to do - for instance, if you are only dealing with real matrices. It is definitely is something a Sage user needs to know, though.

Luckily, it’s possible to restore symbolic constants.

sage: reset('i')
sage: i; i^2
I
-1

>>> from sage.all import *
>>> reset('i')
>>> i; i**Integer(2)
I
-1

reset('i')
i; i^2

sage: type(e)
<class 'sage.symbolic.expression.E'>

>>> from sage.all import *
>>> type(e)
<class 'sage.symbolic.expression.E'>

type(e)

sage: type(pi)
<class 'sage.symbolic.expression.Expression'>

>>> from sage.all import *
>>> type(pi)
<class 'sage.symbolic.expression.Expression'>

type(pi)

Variables are another thing to keep in mind. As mentioned briefly in earlier tutorials, in order to maintain maximum flexibility while not allowing things to happen which shouldn’t, only x is predefined, nothing else.

sage: type(x)
<class 'sage.symbolic.expression.Expression'>

>>> from sage.all import *
>>> type(x)
<class 'sage.symbolic.expression.Expression'>

type(x)

sage: type(y)
Traceback (most recent call last):
...
NameError: name 'y' is not defined

>>> from sage.all import *
>>> type(y)
Traceback (most recent call last):
...
NameError: name 'y' is not defined

type(y)

Warning

There is a way to get around this, but it unleashes a horde of potential misuse. See the cells below if you are interested in this.

sage: automatic_names(True) # not tested
sage: trig_expand((2*x + 4*y + sin(2*theta))^2) # not tested
4*(sin(theta)*cos(theta) + x + 2*y)^2

>>> from sage.all import *
>>> automatic_names(True) # not tested
>>> trig_expand((Integer(2)*x + Integer(4)*y + sin(Integer(2)*theta))**Integer(2)) # not tested
4*(sin(theta)*cos(theta) + x + 2*y)^2

automatic_names(True) # not tested
trig_expand((2*x + 4*y + sin(2*theta))^2) # not tested

This only works in the notebook. Now we’ll turn it off.

sage: automatic_names(False) # not tested

>>> from sage.all import *
>>> automatic_names(False) # not tested

automatic_names(False) # not tested

Another related issue is that a few names are “reserved” by Python/Sage, and which aren’t allowed as variable names.

It’s not surprising that ‘for’ is not allowed, but neither is ‘lambda’ (\(\lambda\))! People often request a workaround for that.

sage: var('lambda')
Traceback (most recent call last):
...
ValueError: The name "lambda" is not a valid Python identifier.

>>> from sage.all import *
>>> var('lambda')
Traceback (most recent call last):
...
ValueError: The name "lambda" is not a valid Python identifier.

var('lambda')

There are lots of ways to get around this. One popular, though annoying, way is this.

sage: var('lambda_')
lambda_

>>> from sage.all import *
>>> var('lambda_')
lambda_

var('lambda_')

sage: lambda_^2-1
lambda_^2 - 1

>>> from sage.all import *
>>> lambda_**Integer(2)-Integer(1)
lambda_^2 - 1

lambda_^2-1

Still, in this one case, showing the expression still shows the Greek letter.

sage: show(lambda_^2-1)

>>> from sage.all import *
>>> show(lambda_**Integer(2)-Integer(1))

show(lambda_^2-1)
\[\lambda^{2} - 1\]

Finally, there is another thing that can happen if you rename things too loosely.

sage: A = matrix(QQ,[[1,2],[3,4]])
sage: B = A
sage: C = copy(A)

>>> from sage.all import *
>>> A = matrix(QQ,[[Integer(1),Integer(2)],[Integer(3),Integer(4)]])
>>> B = A
>>> C = copy(A)

A = matrix(QQ,[[1,2],[3,4]])
B = A
C = copy(A)

This actually has just made B and A refer to the same matrix. B isn’t like A, it is A. The copy command gets around this (though not always).

sage: A[0,0]=987

>>> from sage.all import *
>>> A[Integer(0),Integer(0)]=Integer(987)

A[0,0]=987
sage: show([A,B,C])

>>> from sage.all import *
>>> show([A,B,C])

show([A,B,C])
\[\begin{split}\left[\left(\begin{array}{rr} 987 & 2 \\ 3 & 4 \end{array}\right), \left(\begin{array}{rr} 987 & 2 \\ 3 & 4 \end{array}\right), \left(\begin{array}{rr} 1 & 2 \\ 3 & 4 \end{array}\right)\right]\end{split}\]

This is very subtle if you’ve never programmed before. Suffice it to say that it is safest to let each = sign stand for one thing, and to avoid redundant equals (unlike students at times).

Appendix: Advanced Introductory Topics

There are several things which are useful to know about, but which are not always introduced immediately in programming. We give a few examples here, but they are mainly here to make sure you have seen them so that they are not completely surprising when they come up again.

We saw the “block” structure of Python earlier, with the indentation. This gives the opportunity to introduce conditional statements and comparisons. Here, we just give an example for those who have seen conditionals (“if” clauses) before.

sage: B = matrix([[0,1,0,0],[0,0,1,0],[0,0,0,1],[0,0,0,0]])
sage: for i in range(5): # all integers from 0 to 4, remember
....:     if B^i==0: # We ask if the power is the zero matrix
....:         print(i)
4

>>> from sage.all import *
>>> B = matrix([[Integer(0),Integer(1),Integer(0),Integer(0)],[Integer(0),Integer(0),Integer(1),Integer(0)],[Integer(0),Integer(0),Integer(0),Integer(1)],[Integer(0),Integer(0),Integer(0),Integer(0)]])
>>> for i in range(Integer(5)): # all integers from 0 to 4, remember
...     if B**i==Integer(0): # We ask if the power is the zero matrix
...         print(i)
4

B = matrix([[0,1,0,0],[0,0,1,0],[0,0,0,1],[0,0,0,0]])
for i in range(5): # all integers from 0 to 4, remember
    if B^i==0: # We ask if the power is the zero matrix
        print(i)

We use the double equals sign to test for equality, because = assigns something to a variable name. Notice again that colons and indentation are the primary way that Sage/Python indicate syntax, just as commas and spaces do in English.

Another useful concept is that of a dictionary . This can be thought of as a mathematical mapping from “keys” to “values”. The order is not important and not guaranteed. A dictionary is delimited by curly brackets and correspondence is indicated by colons.

Again, we will just give a small example to illustrate the idea.

What if one wants to specify a matrix using just the nonzero entries? A dictionary is a great way to do this.

This one puts 3 as an entry in the \((2,3)\) spot, for example (remember, this is the third row and fourth column, since we start with zero).

sage: D = {(2,3):3, (4,5):6, (6,0):-3}
sage: C = matrix(D)
sage: C
[ 0  0  0  0  0  0]
[ 0  0  0  0  0  0]
[ 0  0  0  3  0  0]
[ 0  0  0  0  0  0]
[ 0  0  0  0  0  6]
[ 0  0  0  0  0  0]
[-3  0  0  0  0  0]

>>> from sage.all import *
>>> D = {(Integer(2),Integer(3)):Integer(3), (Integer(4),Integer(5)):Integer(6), (Integer(6),Integer(0)):-Integer(3)}
>>> C = matrix(D)
>>> C
[ 0  0  0  0  0  0]
[ 0  0  0  0  0  0]
[ 0  0  0  3  0  0]
[ 0  0  0  0  0  0]
[ 0  0  0  0  0  6]
[ 0  0  0  0  0  0]
[-3  0  0  0  0  0]

D = {(2,3):3, (4,5):6, (6,0):-3}
C = matrix(D)
C

That was a lot easier than inputting the whole matrix!

Finally, although Sage tries to anticipate what you want, sometimes it does matter how you define a given element in Sage.

  • We saw this above with matrices over the rationals versus integers, for instance.

Here’s an example with straight-up numbers.

sage: a = 2
sage: b = 2/1
sage: c = 2.0
sage: d = 2 + 0*I
sage: e = 2.0 + 0.0*I

>>> from sage.all import *
>>> a = Integer(2)
>>> b = Integer(2)/Integer(1)
>>> c = RealNumber('2.0')
>>> d = Integer(2) + Integer(0)*I
>>> e = RealNumber('2.0') + RealNumber('0.0')*I

a = 2
b = 2/1
c = 2.0
d = 2 + 0*I
e = 2.0 + 0.0*I

We will not go in great depth about this, either, but it is worth knowing about. Notice that each of these types of numbers has or does not have \(I=\sqrt{-1}\), decimal points, or division.

sage: parent(a)
Integer Ring
sage: parent(b)
Rational Field
sage: parent(c)
Real Field with 53 bits of precision
sage: parent(d)
Number Field in I with defining polynomial x^2 + 1 with I = 1*I
sage: parent(e)
Complex Field with 53 bits of precision

>>> from sage.all import *
>>> parent(a)
Integer Ring
>>> parent(b)
Rational Field
>>> parent(c)
Real Field with 53 bits of precision
>>> parent(d)
Number Field in I with defining polynomial x^2 + 1 with I = 1*I
>>> parent(e)
Complex Field with 53 bits of precision

parent(a)
parent(b)
parent(c)
parent(d)
parent(e)