TN samacheer kalvi 10th Maths:
Chapter 1 Relations and Functions – Exercise 1.1 introductions
1. What are sets?
A
set is a collection of well-defined objects.
2. Ordered
pair:
A
pair of numbers, written in a particular order, such a number pair is called an
ordered pair of numbers
Example:
(1, 5), (7, 16), (3, 4), (10, 12)
3.
Cartesian Product:
If A and B are two non-empty sets,
then the set of all ordered pairs (a,
b) such that, a ∈ A,
b ∈ B is called the Cartesian Product of A and B, and is denoted
by AxB .Thus, AxB = {(a,b) |a ∈ A,b ∈ B}
(read as A cross B).
A= {1,
2, 3}
B= {a,
b}
AxB={ (1,a),
(1,b)
(2,a),
(2,b)
(3,a),
(3,b) }
B= {a,
b}
A= {1,
2, 3}
BxA={ (a,1),(a,2),(a,3),
(b,1),(b,2),(b,3)};
4. Cartesian
Product of three Sets:
If A, B, C are three non-empty sets
then the cartesian product of three sets is the set of all possible ordered
triplets given by
A×B×C
= {(a,b,c) for all a ∈ A,b ∈ B,c ∈ C}
Let A = {0,1}, B = {0,1}, C = {0,1}
Ax B = {0,1}x{0,1} = {(0,0),(0,1),(1,0),(1,1)}
(AxB)xC = {(0,0),(0,1),(1,0),(1,1)}x{0,1}
= {(0, 0, 0),(0, 0,1),(0, 1, 0),(0, 1,
1),(1, 0, 0),(1, 0, 1)(1, 1, 0),(1, 1, 1)}
5. Important
properties/note:
· A ×
B is
the set of all possible ordered pairs between the elements of A and
B such
that the first coordinate is an element of A and the second coordinate is an
element of B.
· B ×
A is
the set of all possible ordered pairs between the elements of A and
B such
that the first coordinate is an element of B and the second coordinate is an
element of A.
· In
general (a,
b)
≠ (b,
a),
in particular, if a = b, then (a, b)
= (b,
a).
· The
“cartesian product” is also referred as “cross product”.
· In
general A×B ≠ B×A, but, n(A×B) = n(B×A)
· A×B=∅ if and only if A = ∅ or
B = ∅
· If
n(A) = p and n(B) =q then, n(A×B) = pq
6. Types
of numbers:
1. Natural
numbers, N = {1, 2, 3, 4…}
2. Whole
numbers, W= {0, 1, 2, 3,...}
3. Integers,
Z= {…-2,-1, 0, 1, 2, 3,…}
4. Rational
numbers, Q={p/q, | p,q∈Z, q≠0}
5. Prime
numbers, {2, 3, 5, 7, 11, 13,….}
6. Non-prime
numbers {1, 4, 6, 8, 9,….}
7. Even
numbers, {2, 4, 6, 8,….}
8. Odd numbers, {1, 3, 5, 7, 9,….}