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