Lecture 06 Part 1
Sets
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Sets
Sets
The study of sets is called Set Theory [1]. The concept of sets is used in almost all areas of math including algebra, calculus, probability, and geometry. A set is a well-defined collection of objects. By well-defined, we mean that membership to the collection is precise, unambiguous, and with distinguishable limits. For example, "the collection of all vowels in the English alphabet" is a set but, the "collection of all beautiful faces" is not a set because "beautiful faces" is not a well-defined characteristic.
Any member of a set is called an element. For example, since "a" is a vowel, then "a" is an element of the set mentioned above. If a is an element of S, then we write a ∈ S. Otherwise, if a is not an element of S, then we write a ∉ S. Sets are usually denoted with an English capital letter such as "S".
There are two ways of writing out sets:
Roster Notation. In roster notation, the elements of the set are listed.
S = {a, e, i, o, u}
Set Builder Notation. In the set builder notation, the characteristic that defines the collection is specified.
S = {x | x is a vowel in the English alphabet}
Here, x functions as a variable (and not the actual letter x). This is read as "S is the set of all objects x such that x is a vowel in the English alphabet."
Exercise. Complete the table by giving the roster or set builder notation of the set.
Two sets are said to be equal if they have exactly the same elements. That is, A = B if and only if every element of A is also an element of B and at the same time, every element of B is also an element of A. For example,
{a, e, i, o, u} = {x | x is a vowel in the English alphabet}
The way the elements are arranged in the roster notation does not matter. That is,
{a, e, i, o, u} = {u, o, i, e, a} = {e, o, a, u, i}
Each element in the roster notation is said to be unique. This is the same as saying that if the same element is written twice, then both are counted as one.
{a, e, i, o, u} = {a, e, i, o, u, a}
B ⊆ A. This illustration is called a Venn diagram.
If A is a set, then B is said to be a subset of A if and only if every element of B is also an element of A. This is written as B ⊆ A. For example, if A = {a, e, i, o, u} and B = {a, e}, then it is obvious that B ⊆ A. From this definition, it can be realized that every set is a subset of itself (why?). By using the definition of a subset, we could also define set equality alternatively: A = B if and only if A ⊆ B and B ⊆ A.
There is a set which contains all elements of interest. We call this the universal set denoted by U. Since it contains all elements, any set is a subset of the universal set. There is also a set which does not contain any element. This is called the empty set or null set which is denoted by either {} or ∅. It can also be shown that the empty set is a subset of any set.
Set Operations
Set Operations
Like numbers, there are also operations for sets. These are:
union (∪);
intersection (∩);
universal complement (ᶜ); and,
relative complement (-).
The union of two sets A and B is defined by
In other words, the union of two sets is composed of the elements of either or both of these sets. For example, if A = {a, b, c, d} and
B = {c, d, e, f}, then
The intersection of two sets is defined by
In other words, the intersection of two sets is composed of the elements that are common to both sets. For example, if A and B are defined as above, then
For us to obtain the universal complement (or simply, complement), the universal set has to be defined first. The complement of a set A is the set of all elements that do not belong to A. That is,
For example, if U = { a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z} and A = {a, b, c, d}, then,
The complement of A relative to B, is the set of all elements of B that are not in A. In other words,
For example, if A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. Then, A - B = {1, 2}.
Exercise. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 3, 5, 7, 9}, B = {2, 4, 6, 8, 10}, C = {4, 5, 6, 7, 8}. Find
References
References
[1] J. Bagaria. Set Theory. Stanford Encyclopedia of Philosophy. 2019. From https://plato.stanford.edu/entries/set-theory/
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