Fundamentals of Set Theory

This eighteenth article of the mathematical journey through open source, digs into the fundamentals of the set theory through Maxima.

<< Seventeenth Article

Most of you would have definitely heard about the word “set” as in “set theory”, and many of you may already know quite a bit about the sets. Then what’s so special about it. The specialty lies in the whole idea of getting our familiar maths being done by the computer. In this series, we have been mostly picking up familiar maths and then figuring out how that can be done by computer, and that also with no or very little programming knowledge. The same holds true for the sets as well. So, let’s get started with its fundamentals, starting from their creation.

Creation of sets

Just for the beginners of set theory, a set is an unordered collection of distinct items – any item, in any order, but unique. An item is commonly referred to as an element, as well. If an item is contained in a set, it is commonly referred as a member of the set. A set is typically represented by its members enclosed in braces {} and separated by commas. {6, -5, 9, 0}, {dog, cat, donkey, cow, buffalo}, {Kiran, Kasturi, Karan}, {6, horse, sapphire} are some examples. Notice that the first three sets have related items in them, but the last one doesn’t – and that’s perfectly fine. However, if the items in a set have relation(s) or condition(s), the set can also be expressed with that relation(s) or condition(s) mentioned within braces {}. For example, {All human beings younger than 35 years}, {All positive even numbers, All multiples of 3}. In Maxima, we can straight away represent the sets in the first notation, as follows:

$ maxima -q
(%i1) {6, -5, 9, 0};
(%o1)                           {- 5, 0, 6, 9}
(%i2) {dog, cat, donkey, cow, buffalo};
(%o2)                  {buffalo, cat, cow, dog, donkey}
(%i3) {Kiran, Kasturi, Karan};
(%o3)                       {Karan, Kasturi, Kiran}
(%i4) {6, horse, sapphire};
(%o4)                        {6, horse, sapphire}
(%i5) {axe, knife, spear, axe, scissor};
(%o5)                    {axe, knife, scissor, spear}
(%i6) quit();

Note that as the order of items in the set doesn’t matter, Maxima internally keeps them sorted, and hence displayed accordingly, in the above examples. Also note the last example – the duplicates are treated as a single item.

Sets can also be created from ordered lists using setify(). Items of sets could be expressions, but may not be automatically simplified. Check out the following:

$ maxima -q
(%i1) setify([x, y, z]);
(%o1)                              {x, y, z}
(%i2) [x, y, z, x]; /* Ordered list */
(%o2)                            [x, y, z, x]
(%i3) setify([x, y, z, x]);
(%o3)                              {x, y, z}
(%i4) string({x^2 - 1, (x + 1) * (x -1)});
(%o4)                         {(x-1)*(x+1),x^2-1}

string() has been used in %i4, to just have the output on a single line. But the important thing to note is that though the two items of the list are mathematically identical, they have been preserved as two distinct items – and thus not a set in real sense. Such cases can be actually setified by simplifying the individual items of the set using corresponding simplification function, e.g. rat() for rational expressions. And operating any function on every item of a set can be achieved using map(). Here’s an to example to set all those straight, continuing from the above one:

(%i5) string(map(rat, {x^2 - 1, (x + 1) * (x -1)}));
(%o5)                               {x^2-1}
(%i6) string(rat((x + 1) * (x -1)));
(%o6)                                x^2-1
(%i7) quit();

%i6 and %o6 above is just to demonstrate how rat() works. I know, you are still wondering what this wierd map() is and how it is working. So, here are a few more examples for the same:

$ maxima -q
(%i1) trigreduce(2 * sin(x) * cos(x));
(%o1)                              sin(2 x)
(%i2) {sin(2 * x), 2 * sin(x) * cos(x)};  /* Identical items */
(%o2)                     {2 cos(x) sin(x), sin(2 x)}
(%i3) map(trigreduce, {sin(2 * x), 2 * sin(x) * cos(x)});
(%o3)                             {sin(2 x)}
(%i4) string({apple / fruit + mango / fruit, (apple + mango) / fruit});
(%o4)            {(mango+apple)/fruit,mango/fruit+apple/fruit}
(%i5) string(map(rat, {apple / fruit + mango / fruit, (apple + mango) / fruit}));
(%o5)                        {(mango+apple)/fruit}
(%i6) quit();

In fact, the power of map() lies in its ability that it can take a function created on the fly, meaning using the lambda notation. Here goes few examples to demonstrate the lamda() first, and then map() using lambda():

$ maxima -q
(%i1) f: lambda([x], x^3)$
(%i2) f(5);
(%o2)                                 125
(%i3) lambda([x], x^3)(5);
(%o3)                                 125
(%i4) lambda([x, y], x+y)(4, 6);
(%o4)                                 10
(%i5) map(f, {0, 1, 2, 3});
(%o5)                            {0, 1, 8, 27}
(%i6) map(lambda([x], x^3), {0, 1, 2, 3});
(%o6)                            {0, 1, 8, 27}
(%i7) map(lambda([x, y], x+y), {a}, {3});
(%o7)                               {a + 3}
(%i8) map(lambda([x], x^4), {-2, -1, 0, 1, 2});
(%o8)                            {0, 1, 16}
(%i9) map(g, {-2, -1, 0, 1, 2});
(%o9)                {g(- 2), g(- 1), g(0), g(1), g(2)}
(%i10) quit();

lambda() takes two arguments. First a list of arguments of the function being defined, and second the expression for the return value of the function using those arguments. %i1 defines a function f with one argument, returning its cube. %i2 calls f(). However, the whole point of using lambda is using it without defining an explicit function like f(). So, %i3 & %i4 demonstrate exactly that. %i6, %i7, %i8 shows using lambda() with map(). Note the elimination of duplicates in %o8. %i9 is an another example of map().

Basic set operations

Enough with varieties of set creation. Let’s do some set operations. For the starters: union of sets is defined as a set with items of all the sets; intersection of sets is defined as a set with items common in all the sets; difference of two sets is defined as a set with items from the first set, not in the second set. And here goes the demonstration:

$ maxima -q
(%i1) union({1, 2}, {1, 3, 4}, {1, 2, 6, 7});
(%o1)                         {1, 2, 3, 4, 6, 7}
(%i2) intersection({1, 2}, {1, 3, 4}, {1, 2, 6, 7});
(%o2)                                 {1}
(%i3) setdifference({1, 2}, {1, 3, 4});
(%o3)                                 {2}
(%i4) quit();

Other basic set operations provided by Maxima are:-

  • cardinality() – returns the number of distinct items in a set
  • elementp() – checks for an item to be a member of a set
  • emptyp() – checks for the emptiness of a set
  • setequalp() – compares two sets for equality
  • disjointp() – checks for no common items in two sets
  • subsetp() – checks for the first set to be a subset of the second set

 

The following walk through demonstrates all of these:

$ maxima -q
(%i1) S1: {}$
(%i2) S2: {1, 2, 3}$
(%i3) S3: {3, 1, 5-3}$ /* Same as S2 */
(%i4) S4: {a, b, c}$
(%i5) S5: {2, 1, 2}$
(%i6) cardinality(S1);
(%o6)                                  0
(%i7) cardinality(S2);
(%o7)                                  3
(%i8) cardinality(S3);
(%o8)                                  3
(%i9) cardinality(S4);
(%o9)                                  3
(%i10) cardinality(S5);
(%o10)                                 2
(%i11) elementp(b, S3);
(%o11)                               false
(%i12) elementp(b, S4);
(%o12)                               true
(%i13) emptyp(S1);
(%o13)                               true
(%i14) emptyp(S2);
(%o14)                               false
(%i15) setequalp(S1, S2);
(%o15)                               false
(%i16) setequalp(S2, S3);
(%o16)                               true
(%i17) disjointp(S1, S2);
(%o17)                               true
(%i18) disjointp(S2, S3);
(%o18)                               false
(%i19) disjointp(S3, S4);
(%o19)                               true
(%i20) disjointp(S3, S5);
(%o20)                               false
(%i21) subsetp(S1, S2);
(%o21)                               true
(%i22) subsetp(S2, S3);
(%o22)                               true
(%i23) subsetp(S3, S2);
(%o23)                               true
(%i24) subsetp(S3, S4);
(%o24)                               false
(%i25) subsetp(S5, S3);
(%o25)                               true
(%i26) subsetp(S3, S5);
(%o26)                               false
(%i27) quit();

Playing with set elements

After clearing the fundamentals, mostly through numerical examples, it is now time to have some fun with symbol substitution of Maxima. And here’s some playing around:

$ maxima -q
(%i1) S: {a, b, c, a};
(%o1)                             {a, b, c}
(%i2) S: {a+b, b+c, c+d, d+a};
(%o2)                   {b + a, c + b, d + a, d + c}
(%i3) subst(a=c, S);
(%o3)                          {c + b, d + c}
(%i4) subst([a=c, b=d], S);
(%o4)                              {d + c}
(%i5) subst([a=c, b=d, c=-d], S);
(%o5)                                {0}
(%i6) subst([a=1, b=2, c=-3], S);
(%o6)                       {- 1, 3, d - 3, d + 1}
(%i7) T: {S, {S}};
(%o7)   {{b + a, c + b, d + a, d + c}, {{b + a, c + b, d + a, d + c}}}
(%i8) subst([a=c, b=d, c=-d], T);
(%o8)                            {{0}, {{0}}}
(%i9) subst([a=1, b=2, c=-3], T);
(%o9)         {{- 1, 3, d - 3, d + 1}, {{- 1, 3, d - 3, d + 1}}}
(%i10) quit();

Nineteenth Article >>

Anil Kumar Pugalia Anil Kumar Pugalia (105 Posts)

The author is a hobbyist in open source hardware and software, with a passion for mathematics, and philosopher in thoughts. A gold medallist from the Indian Institute of Science, Linux, mathematics and knowledge sharing are few of his passions. He experiments with Linux and embedded systems to share his learnings through his weekend workshops. Learn more about him and his experiments at https://sysplay.in.


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Anil Kumar Pugalia

About Anil Kumar Pugalia

The author is a hobbyist in open source hardware and software, with a passion for mathematics, and philosopher in thoughts. A gold medallist from the Indian Institute of Science, Linux, mathematics and knowledge sharing are few of his passions. He experiments with Linux and embedded systems to share his learnings through his weekend workshops. Learn more about him and his experiments at https://sysplay.in.

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