1. Sets --- A set is a collection of objects. Sets are denoted by A, B, C, … --- The objects in the set are called the elements of the set. The elements are denoted by a, b, c, …

2. Sets --- If a is an element on a set A we write a A if a is not in A we write a A. --- The elements in a set X can be written in the form X = {.......,.......,.......}. --- If X has no elements we write X = {} or  and call it the empty set.

3. Sets --- X  Y denotes the intersection of two sets X and Y and it is the set which contains all the elements belonging to X and Y. --- X  Y denotes the union of two sets X and Y and it is the set which contains all the elements belonging to X or to Y. --- Sets can be finite, infinite or empty.

4. Exercises: (1) Identify the following sets (a) S = {students that can fly} (b) B the set of positive integers less than or equal to 3. (2) If A = {1,3,5,7,9}, B = {1,2,3,4,5} and C ={5,6,7,8,9} find i) A ∩ B ii) A U C iii) A U B U C iv) A ∩ B ∩ C

5. Operations on Real Numbers: --- The set of Real Numbers is denoted by the symbol R : --- The operations on real number are addition, subtraction, multiplication and division.

6. Exercises: (1) Add 5 to 6. (2) Subtract 3 from 7. (3) Multiply 3 by 8. (4) Divide 30 by 10. (5) Multiply 20 by 5 and then subtract 43..

7. The set of Real numbers has important subsets: The natural numbers: 1, 2, 3, … is denoted by N The integers: …, -2, -1, 0, 1, 2,... is denoted by Z The rational numbers: the real numbers of the form a/b where a, b are integers and b # 0 is denoted by Q.

8. Excercises: Circle the right answer: (1) The real number 3/5 is (a) an integer (b) not a rational number (c) a natural number (d) not an integer (2) If you add an integer to a rational number you always get: (a) an integer (b) a rational number (c) a natural number (d) a negative real number. (3) Two integers, the ratio between them is 3 : 7 and if substracted 5 from each term, the ratio between each of them becomes 1 : 3. Find the two numbers?