Sets

1. Notation

Sets will usually be denoted by capital letters;

A, B, X, Y,......

Lower case letters will usually represent the elements in our sets:

Lets take as an example; if we define a particular set by actually listing its members, for example, let A consist of numbers 1,3,7, and 10, then we write

A={1,3,7,10}

That is, the elements are separated by commas and enclosed in brackets {}.

We call this the tabular form of a set

Now, try your hand on this

Exercise 1.1

State in words and then write in tabular form

1. A = {x}│x ² = 4²}

2. B = {x}│x – 2 = 5}

3. C = {x}│x is positive, x is negative}

4. D = {x}│x is a letter in the word “correct”}

Solution:

1. It reads “A is the set of x such that x squared equals four”. The only numbers which when squared give four are 2 and -2. Hence A = {2, -2}

2. It reads “B is the set of x such that x minus 2 equals 5”. The only

solution is 7; hence B = {7}

3. It read “C is the set of x such that x is positive and x is negative”. There is no number which is both positive and negative; hence C is empty, that is, C=Ø

4. It reads “D is the set of x such that x is letter in the work ‘correct’. The indicated letters a re c,o,r,e and t; thus D = {c,o,r,e,t}

But if we define a particular set by stating properties which its elements must satisfy, for example, let B be the set of all even numbers, then we use a letter, usually x, to represent an arbitrary element and we write:

B = {x│x is even}

Which reads “B is the set of numbers x such that x is even”. We call this the

set builders form of a set. Notice that the vertical line “│” is read “such as”.

In order to illustrate the use of the above notations, we rewrite the sets in

examples 1.1-1.10. We denote the sets by A 1, A 2, ..... A10 respectively.

Example 2.1: A1 = {0, 2, 4, 6, 8}

Example 2.2: A2 = {x│x²+2x + 1 = 0}

Example 2.3: A3 = {a, e, i, o, u}

Example 2.4: A4 = {x │x is a person living on the earth}

Example 2.5: A5 = {Tom, Dick, Harry}

Example 2.6: A6 = {x │x is a student and x is absent from school}

Example 2.7: A7 = {England, France, Denmark}

Example 2.8: A8 = {x│x is a capital city and x is in Nigeria}

Example 2.9: A9 = {1, 3, 7, 10}

Example 2.10: A10 = { x│x is a river and x is in Nigeria} It is easy as that!

Exercise 1.2

Write These Sets in a Set-Builder Form

1. Let A consist of the letters a, b, c, d and e

2. Let B = {2, 4, 6, 8........}

3. Let C consist of the countries in the United Nations

4. Let D = {3}

5. Let E be the Heads of State Obasanjo, Yaradua and Jonathan

Solution

1.A={x │x appears before f in the alphabet} = {xx is one of the first letters in the alphabet}

2. B = {x │x is even and positive}

3. C = {x │x is a country, x is in the United Nations}

4. D = {x │x – 2 = 1} = {x│2x = 6}

5. E = {x │x was Head of state after Abacha}

If an object x is a member of a set A, i.e., A contains x as one of its\elements, then we write: x∈A

Which can be read “x belongs to A” or ‘x is in A”. If, on the other hand, an object x is not a member of a set A, i.e A does not contain x as one of its elements, then we write; x∉A

It is a common custom in mathematics to put a vertical line “” or “ \ ” through a symbol to indicate the opposite or negative meaning of the symbol.

Example 3:1: Let A = {a, e, i o, u}. Then a∈A, b∉A, f∉A.

Example 3.2: Let B = {x< x is even}. Then 3∉B, 6∈B, 11∉B,14∈B

2. 2. Finite & Infinite Sets

Sets can be finite or infinite. Intuitively, a set is finite if it consists of a specific number of different elements, i.e. if in counting the different members of the

set the counting process can come to an end. Otherwise a set is infinite. Lets look at some examples.

Example 4:1: Let M be the set of the days of the week. The M is finite

Example 4:2: Let N = {0,2,4,6,8........}. Then N is infinite

Example 4:3: Let P = {x< x is a river on the earth}. Although it maybe difficult to count the number of rivers in the world, P is still a finite set.

Exercise 1.3: Which sets are finite?

1. The months of the year

2. {1, 2, 3, ......... 99, 100}

3. The people living on the earth

4. {x | x is even}

5. {1, 2, 3,........}

Solution:

The first three sets are finite. Although physically it might be impossible to count the number of people on the earth, the set is still finite. The last two sets are infinite. If we ever try to count the even numbers, we would never come to the end.

3. Equality of Sets

Set A is equal to set B if they both have the same members, i.e if every element which belongs to A also belongs to B and if every element which belongs to B also belongs to A. We denote the equality of sets A and B by:

A = B

Example 5.1 Let A = {1, 2, 3, 4} and B = {3, 1, 4, 2}. Then A = B,

that is {1,2,3,4} = {3,1,4,2}, since each of the elements 1,2,3 and 4 of A belongs to B and each of the elements 3,1,4 and 2 of B belongs to A. Note therefore that a set does not change if its elements are rearranged.

Example 5.3 Let E={x | x²–3x = -2}, F={2,1} and G ={1,2,2, 1},

Then E= F= G

4. Null Set

It is convenient to introduce the concept of the empty set, that is, a set which contains no elements. This set is sometimes called the null set.

We say that such a set is void or empty, and we denote its symbol ∅

Example 6.1: Let A be the set of people in the world who are older than 200years. According to known statistics A is the null set.

Example 6.2: Let B = {x | x²= 4, x is odd}, Then B is the empty set.

5. SUBSETS

If every element in a set A is also a member of a set B, then A is called subset of B.

More specifically, A is a subset of B if x∈A implies x∈B. We denote this relationship by writing; A⊂B, which can also be read “A is contained in B”.

Example 7.1

The set C = {1,3,5} is a subset of D = {5,4,3,2,1}, since each number 1, 3 and 5 belonging to C also belongs to D.

Example 7.2

The set E = {2,4,6} is a subset of F = {6,2,4}, since each number 2,4, and 6 belonging to E also belongs to F. Note, in particular, that E = F. In a similar manner it can be shown that every set is a subset of itself.

Example 7.3

Let G = {x | x is even}, i.e. G = {2,4,6}, and let F = {x | x is a positive power of 2}, i.e. let F = {2,4,8,16.....} Then F⊂G, i.e. F is contained in G.

With the above definition of a subset, we are able to restate the definition of the equality of two sets.

Two set A and B are equal, i.e. A = B, if an only if A⊂B and B⊂A. If A is a subset of B, then we can also write:

B⊃A

Which reads “B is a superset of A” or “B contains A”. Furthermore, we write:

A⊄B

if A is not a subset of B.

Conclusively, we state:

1. The null set ∅ is considered to be a subset of every set

2. If A is not a subset of B, that is, if A⊄B, then there is at least one element in A that is not a member of B.

6. Proper Subsets

Since every set A is a subset of itself, we call B a proper subset of A if, first, is a subset of A and secondly, if B is not equal to A. More briefly, B is a proper subset of A if:

B⊂A and B≠A

In some books “B is a subset of A” is denoted by

B⊆A

and B” is a proper subset of A” is denoted by

B⊂A

We will continue to use the previous notation in which we do not distinguished between a subset and a proper subset.

7. Comparability

Two sets A and B are said to be comparable if:

A⊂B or B⊂A;

That is, if one of the sets is a subset of the other set. Moreover, two sets A and B are said to be not comparable if:

A⊄B and B⊄A

Note that if A is not comparable to B then there is an element in A.which is not in B and ... also, there is an element in B which is not in A.

Example 8.1: Let A = {a,b} and B = {a,b,c}. The A is comparable to B, since

A is a subset of B.

Example 8.2: Let R – {a,b} and S = {b,c,d}. Then R and S are not comparable, since a∈R and a∉S and c∉R

In mathematics, many statements can be proven to be true by the use of previous assumptions and definitions. In fact, the essence of mathematics consists of theorems and their proofs. We now proof our first

Theorem 1.1: If A is a subset of B and B is a subset of C then A is a subset of

C, that is,

A⊂B and B⊂C implies A⊂B

Proof: (Notice that we must show that any element in A is also an element in C). Let x be an element of A, that is, let x∈A. Since A is a subset of B, x also belongs to B, that is, x∈B. But by hypothesis, B⊂C; hence every element of B, which includes x, is a number of C. We have shown that x∈A implies x∈C.

Accordingly, by definition, A⊂C.

8. Sets of Sets

It sometimes will happen that the object of a set are sets themselves; for example, the set of all subsets of A. In order to avoid saying “set of sets”, it is common practice to say “family of sets” or “class of sets”. Under the circumstances, and in order to avoid confusion, we sometimes will let script

letters A, B,..........

Denote families, or classes, of sets since capital letters already denote their elements.

Example 9.1: In geometry we usually say “a family of lines” or “a family of curves” since lines and curves are themselves sets of points.

Example 9.2: The set {{2,3}, {2}, {5,6}} is a family of sets. Its members are the sets {2,3}, {2} and {5,6}.

Theoretically, it is possible that a set has some members, which are sets themselves and some members which are not sets, although in any application of the theory of sets this case arises infrequently

Example 9.3: Let A = {2, {1,3}, 4, {2,5}}. Then A is not a family of sets; here some elements of A are sets and some are not.

9. Universal Set

In any application of the theory of sets, all the sets under investigation will likely be subsets of a fixed set. We call this set the universal set or universe of discourse. We denote this set by U.

Example 10.1: In plane geometry, the universal set consists of all the points in the plane.

Example 10.2: In human population studies, the universal set consists of all the people in the world.

10. Power Set

The family of all the subsets of any set S is called the power set of S.

We denote the power set of S by: 2S

Example 11.1: Let M = {a,b} Then 2

M = {{a, b}, {a}, {b}, ∅}

Example 11.2: Let T = {4,7,8} then 2

T = {T, {4,7}, {4,8}, {7,8}, {4}, {7},

{8}, ∅}

If a set S is finite, say S has n elements, then the power set of S can be shown to have 2n elements. This is one reason why the class of subsets of S is called

the power set of S and is denoted by 2 S.

11. Disjoint Sets

If sets A and B have no elements in common, i.e if no element of A is in B and no element of B is in A, then we say that A and B are disjoint

Example 12.1: Let A = {1,3,7,8} and B = {2,4,7,9}, Then A and B are not disjoint since 7 is in both sets, i.e 7∈A and 7∈B

Example 12.2: Let A be the positive numbers and let B be the negative numbers. Then A and B are disjoint since no number is both positive and negative.

Example 12.3: Let E = {x, y, z} and F = {r, s, t}, Then E and F are disjoint.

12. VENN-EULER DIAGRAMS

A simple and instructive way of illustrating the relationships between sets is in the use of the so-called Ven-Euler diagrams or, simply, 

Venn diagrams. Here we represent a set by a simple plane area, usually bounded by a circle.

Example 13.1: Suppose A⊂B and, say, A≠B, then A and B can be described by either diagram

13. AXIOMATIC DEVELOPMENT OF SET THEORY

In an axiomatic development of a branch of mathematics, one begins with:

1. Undefined terms

2. Undefined relations

3. Axioms relating the undefined terms and undefined relations. Then, one develops theorems based upon the axioms and definitions Example 14:1:

In an axiomatic development of Plane Euclidean geometry

1. “Points” and “lines” are undefined terms

2. “Points on a line” or, equivalent, “line contain a point” is an undefined relation

3. Two of the axioms are:

Axiom 1: Two different points are on one and only one line

Axiom 2: Two different lines cannot contain more than one point in common

In an axiomatic development of set theory:

1. “Element’ and “set” are undefined terms

2. “Element belongs to a set” is undefined relation

3. Two of the axioms are

Axiom of Extension: Two sets A and B are equal if and only if every element in A belongs to B and every element in B belongs to A.

Axiom of Specification: Let P(x) be any statement and let A be any set. Then there exists a set:

B= {aa ∈ A, P (a) is true}

Here, P(x) is a sentence in one variable for which P(a) is true or false for any a∈A. for example P(x) could be the sentence “x² = 4” or “x is a member of the United Nations”

14. SUMMARY

A summary of the basic concept of set theory is as follows:

 A set is any well-defined list, collection, or class of objects.

 Given a set A with elements 1,3,5,7 the tabular form of representing this set is A = {1, 3, 5, 7}.

 The set-builder form of the same set is A = {x| x = 2n + 1,0 ≤n≤3}

 Given the set N = {2,4,6,8,....} then N is said to be infinite, since the counting process of its elements will never come to an end, otherwise it is finite

 Two sets of A and B are said to be equal if they both have the same elements, written A = B

 The null set,∅ , contains no elements and is a subset of every set

 The set A is a subset of another set B, written A⊂B, if every element of A is also an element of B, i.e. for every x∈A then x∈B

 If B⊂A and B≠A, then B is a proper subset of A

 Two sets A and B are comparable if A⊂B and B⊂A

 The power set 2² of any set S is the family of all the subsets of S

 Two sets A and B are said to be disjoint if they do not have any element in common, i.e their intersection is a null set.