The concept of sets serves as a fundamental part of mathematics. The concepts of relations and functions are defined using sets. Knowledge of sets is required for the study of geometry, sequences, probability etc. It is very important to have knowledge of sets relations and functions for JEE exam. 2 or 3 questions can be expected from this chapter for JEE exam.
Sets
We often speak about a collection of objects such as bunch of flowers, football team, students of a class. We come across such collections in mathematics also. For example, prime numbers, even numbers, etc. These collections are well defined objects. A well defined collection of objects which are distinct from each other is known as a set. Sets are denoted using capital letters A, B, C etc. Elements of a set are represented by small letters a, b, c etc. If x is an element of set X, we say that “x belongs to X”. The phrase belongs to is denoted by Greek symbol epsilon, e .The number of elements present in a set is the cardinality of a set.
There are 2 methods for representation of sets. They are
(i) Roaster or tabular form
(ii)Set builder form
In Roaster form, all elements of a set are listed, the elements are enclosed within braces { } and elements are separated by commas. For example, set of prime numbers less than 10 is denoted in roaster form as {2, 3, 5, 7}.
In Set builder form we describe element of a set by using a symbol a, b, x, y etc which is followed by a colon “:”. After the colon symbol, we write the property possessed by elements of the set. The whole description is then enclosed within braces. For example, the set A = {a : a is an odd number and 4 < a < 10}. This is read as set of all a such that a is an odd number and a lies between 4 and 10. So the numbers 5, 7 and 9 are the elements of set A.
Cartesian product of sets
If A and B are two non empty sets, then the Cartesian product is the set of all ordered pairs from A and B.
i.e., A × B = { (a,b) : a e A and b e B}
If A or B is a null set, then A× B will be a null set.
Relations
A relation R from a non empty set P to a non empty set Q is a subset of the Cartesian product P×Q. This subset is formed by describing a relationship between the first element and the second element of the ordered pair in P×Q. The second element is known as image of first element. The set of all the first elements in an ordered pair is called domain of the relation R. The set of all second elements is called the range of the relation R.
Functions
A relation f from a set P to a set Q is said to be a function if every element of P has one and only one image in Q. A function from P to Q is denoted by f: P®Q
Truth tables and Logical statements
To perform logical operations, truth tables are used. Truth values are true and false denoted by T and F. In Boolean algebra,truth table shows the truth value of a statement for each possible combination of truth values of component statements.For more information about previous year JEE questions and Truth Tables and Logical Statements you may visit BYJUS.
