Exercise
1.1 Logic
The dictionary meaning of logic is the science of reasoning. In mathematics, we deal with various theorems and formulae. The different procedures used in giving the proofs of various theorems and formulae are based on sound reasoning. The study of such procedures based on sound reasoning is known as a logic.
In logic, we use symbols for words, statements and their relations to get the required result. Hence, logic is known as mathematical logic or symbolic logic.
Exercise 1.1
1.2 Statements
An assertion expressed in words or symbols, which is either true or false but not both at the same time, is known as a statement. Some examples of statements are given below:
i) Water is essential for health.
ii) 2+4=6
iii) A quadrilateral has three sides.
(i), (ii) and (iii) are statements as (i) and (ii) are true but (iii) is false.
1.3 Truth value and truth table
A truth or the falsity of a statement is known as its truth value. T or F is the truth value of a statement according as it is true or false. A table presenting the truth values of the component statements together with the truth values of their compound statement, is known as the truth table.
1.4 Logical Connectives
Compound statements are made from the simple statements by using the words or phrase like “and”, “or”, “If …… then” and “If and only if” and they are known as logical connectives or simply connectives.
1.5 Sets
The word “set” is known to carry the same meaning as the words collection, class and aggregate. However a set may be thought of as a well-defined list or collection of material objects such as books and pens or conceptual objects such as numbers and points. Each object of a set is called an element or member of the set.
1.6 Theorems based on Set Operation
Combinations of various relations and operations defined on sets give rise to a number of interesting and useful results. Some of them are so fundamental that they are considered as the basic laws of set algebra. We discuss them under the following four categories.
a) Properties of Inclusion and Equality Relations
b) Properties of Unions
c) Properties of Intersections
d) Miscellaneous Properties
1.7 Real Number System
The union of the set of rational and irrational numbers is known as the set of real numbers. So, by the set of real numbers, we have the set of natural numbers, the set of integers, the set of rational numbers and the set of irrational numbers.
The set of real numbers are denoted by R. Thus the set of rational and irrational numbers taken together, form a new system of numbers known as the real number system.
1.8 Representation of a number in a real line
The set of real numbers can be beautifully represented by means of the points on a straight line which is called a real line.
1.9 Interval
Let a and b be two numbers on the real line. Then the set of points on the real line between a and b is known as an interval, a and b are known as the end points of the interval. An interval is denoted by 1. An interval may or may not include the end points. So, we get the following four different types of intervals.
i) Open-interval
ii) Closed interval
iii) Left open interval
iv) Right open interval
1.10 Absolute Value
Let x denote any real number. The absolute value (or modulus or numerical value) of x. written as |x|, is a non-negative real number defined by ;
|x|= [x, if x >= 0]
|x|= [- x, if x < 0]
