Class 9 general math chapter 2 exercise 2.1 solution
Set: A set is defined as a well-defined collection or assembly of objects from the real or conceptual world. Typically, sets are denoted by uppercase English letters such as A, B, C, … , X, Y, Z.
Each object or member of a set is called an element of the set. For example, if B = {a, b}, then a and b are elements of set B.
Methods of Representing a Set: A set can be represented primarily by two methods:
1. Roster (Listing) Method
2. Set-builder Method
1. Roster Method: In this method, all elements of the set are explicitly listed and enclosed within curly braces { }. If there are multiple elements, they are separated by commas. For example: A = {a, b}, B = {2, 4, 6}, C = {Niloy, Tisha, Shuvra}, etc.
2. Set-builder Method: In this method, all elements of the set are not explicitly listed; instead, a common property that defines the elements is specified. For example: A = {x : x is an odd natural number}, B = {x : x is one of the top five students in grade nine}, etc.
Types of Sets:
Finite Set: A set in which the number of elements can be counted is called a finite set.
Infinite Set: A set in which the number of elements cannot be counted is called an infinite set.
Empty Set: A set that contains no elements is called an empty set. The empty set is denoted by Φ.
Venn Diagram: John Venn (1834-1883) introduced a method of representing sets using diagrams. In this method, the sets under consideration are depicted using various geometric shapes in a plane, such as rectangular, circular, and triangular regions. These diagrams are known as Venn diagrams, named after John Venn.
Subset: Any set that can be formed from the elements of a given set is called a subset of that set
Annual exam preparation for class nine
Proper Subset: If B is a subset of A and A has at least one element that is not in B, then B is called a proper subset of A, and it is written as B ⊆ A. For example: A = {3, 4, 5, 6} and B = {3, 5} are two sets.
Equality of Sets: Two sets are considered equal if they contain the same elements. For example: A = {3, 5, 7} and B = {5, 3, 7} are equal sets, written as A = B.
Difference of Sets: The set that is formed by removing the elements of one set from another set is called the difference of sets.
Universal Set: In practical discussions, all sets under consideration are subsets of a particular set, called the universal set.
For example: If A = {x, y} is a subset of B = {x, y, z}, then B is the universal set relative to A.
Complement of a Set: If U is the universal set and A is a subset of U, then the set containing all elements not in A is called the complement of A. The complement of A is denoted by Ac or A´. Mathematically, Ac = U \ A.
Union of Sets: A set formed by combining all elements of two or more sets is called the union of the sets.
Intersection of Sets: A set formed by the common elements of two or more sets is called the intersection of the sets. Suppose A and B are two sets. The intersection of A and B is denoted by A ∩ B and is read as “A intersection B.” In set-builder notation, A ∩ B = {x : x ∈ A and x ∈ B}.
Disjoint Sets: If two sets have no elements in common, they are called disjoint sets.
Power Set: The power set of a set A is denoted by P(A).
Ordered Pair: An ordered arrangement of two elements, where one is designated as the first position and the other as the second, is called an ordered pair.
Cartesian Product: The set of all ordered pairs formed by the elements of any two sets A and B is called their Cartesian product.
Relation: If A and B are two sets, then a non-empty subset R of the Cartesian product A × B is called a relation from set A to set B.
Function: If two variables x and y are related in such a way that for any value of x, there is only one corresponding value of y, then y is called a function of x.
Domain and Range: For any relation, the set of the first elements in its ordered pairs is called its domain, and the set of the second elements is called its range. Suppose R is a relation from set A to set B, meaning R ⊆ A × B. The set of the first elements in the ordered pairs of R will be the domain of R, and the set of the second elements will be the range of R. The domain of R is denoted by dom R, and the range by range R.
Graph of a Function: The visual representation of a function is called its graph. Graphs play an essential role in clarifying the concept of functions. Two perpendicular intersecting straight lines are called coordinate axes, and the point where they intersect is called the origin.
Perpendicular Axes: In a plane, two straight lines XOX´ and YOY´ are drawn intersecting each other perpendicularly. The horizontal line XOX´ is called the x-axis, the vertical line YOY´ is called the y-axis, and their intersection point O is called the origin.
Coordinates: The signed numbers representing the perpendicular distances from any point in the plane to the axes are called the coordinates of that point.
Creative Question – 1
Given:
ƒ(x) = x² + 4x + 3
A = {x ∈ Ι : x is an odd number and x < 6}
B = {x ∈ Ι : x is a divisor of 21}
C = {x ∈ Ι : x is a multiple of 7 and x < 35}
Solve the following:
a. Find the value of ƒ(−1). 2
b. Show that if the number of elements in A is n, then the number of elements in P(A) is 2ⁿ. 4
c. Prove that A × (B ∩ C) = (A × B) ∩ (A × C). 4
Creative Question Solution:
(a)Given, ƒ(x) = x² + 4x + 3
So,
ƒ(−1) = (−1)² + 4 × (−1) + 3
= 1 − 4 + 3
= 0 (Answer)
**(b)** Given, A = {x ∈ Ι : x is an odd number and x < 6}
So, A = {1, 3, 5}
Subsets of A: {1}, {3}, {5}, {1, 3}, {1, 5}, {3, 5}, {1, 3, 5}, and Æ (empty set)
Thus, P(A) = {{1}, {3}, {5}, {1, 3}, {1, 5}, {3, 5}, {1, 3, 5}, Æ}
The number of elements in set A is 3, and the number of elements in its power set P(A) is 8, which is equal to 2³.
Thus, if the number of elements in set A is n, the number of elements in P(A) is 2ⁿ, as shown.
(c) Given,
B = {x ∈ Ι : x is a divisor of 21}
= {1, 3, 7, 21}
and
C = {x ∈ Ι : x is a multiple of 7 and x < 35}
= {7, 14, 21, 28}
Now,
B ∩ C = {1, 3, 7, 21} ∩ {7, 14, 21, 28} = {7, 21}
Left Side:
A × (B ∩ C) = {1, 3, 5} × {7, 21}
= {(1, 7), (1, 21), (3, 7), (3, 21), (5, 7), (5, 21)}
Right Side:
A × B = {1, 3, 5} × {1, 3, 7, 21}
= {(1, 1), (1, 3), (1, 7), (1, 21), (3, 1), (3, 3), (3, 7), (3, 21), (5, 1), (5, 3), (5, 7), (5, 21)}
A × C = {1, 3, 5} × {7, 14, 21, 28}
= {(1, 7), (1, 14), (1, 21), (1, 28), (3, 7), (3, 14), (3, 21), (3, 28), (5, 7), (5, 14), (5, 21), (5, 28)}
(A × B) ∩ (A × C) = {(1, 7), (1, 21), (3, 7), (3, 21), (5, 7), (5, 21)}
Therefore, A × (B ∩ C) = (A × B) ∩ (A × C)
Question -2
g(x) = \frac{3x + 1}{3x – 1} and h(t) = \frac{t^4 + t^2 + 1}{t^2} are two algebraic expressions.
a. Find the values of g(0) and h(1). 2
b. Find the value of \frac{g\frac({1}{x}) +1}{g\frac({1}{x}) – 1}. 4
c. Prove that h(t^2) = h(\frac{1}{t^2}). 4
Solution for Question 2
(a) Given, g(x) = \frac{3x + 1}{3x – 1}
So, g(0) = \frac{3.0 + 1}{3.0 – 1}
= \frac{0 + 1}{0 – 1}
= \frac{1}{– 1}
= -1 (Ans.)
and h(t) = \frac{t^4 + t^2 + 1}{t^2}
h(1) = \frac{1^4 + 1^2 + 1}{1^2}
= \frac{1 + 1 + 1}{1}
= \frac{3}{1}
= 3
(b) Similar to Example 24 from Exercise 2.2 in the textbook.
(c) Given, h(t) = \frac{t^4 + t^2 + 1}{t^2}
So, h (t^2) = \frac{(t^2)^4 + (t^2)^2 + 1}{(t^2)^2}
= \frac{t^8 + t^4 + 1}{t^4}
and h(\frac{1}{t^2}) = \frac{(\frac{1}{t^2})^4 + (\frac{1}{t^2})^2 + 1}{(\frac{1}{t^2})^2}
= \frac{\frac{1}{t^8} + \frac{1}{t^4} + 1}{\frac{1}{t^4}}
= \frac{\frac{1 + t^4 + t^8}{t^8}}{\frac{1}{t^4}}
= \frac{1 + t^4 + t^8}{t^4}
Therefore, h(t^2) = h(\frac{1}{t^2}) (Proved)
