10 maths Chapter 1 Course

Set Theory & Cartesian Product - Key Concepts

2) Order Pairs (வரிசைச் சோடிகள்)

Such a number pair is called ordered pairs of numbers. For example: \((a, b)\).

3) Cartesian Product (கார்டீசியன் பெருக்கல்)

\(A\) and \(B\) are two non-empty sets. Order pairs \((a, b)\) such that \(a \in A\), \(b \in B\).

So it is denoted as \(A \times B\).

Suppose if \(A \times \phi = \phi\) (That is important)

4) Elements Position in Cartesian Product

If \(A \times B\), set of order pairs. \(A\) is an first element, \(B\) is a second element.

For Example 1:
If they given \(A \times B = \{(1,2), (2,2), (3,2)\}\), then find \(A\) and \(B\).
So, we can write:
\(A = \{1, 2, 3\}\)
\(B = \{2\}\)

For Example 2:
If they given \(B \times A = \{(2,1), (2,2), (2,3)\}\), then the first element is \(B\), so we find:
\(B = \{2\}\)
\(A = \{1, 2, 3\}\)

5) Commutative Property & Number of Elements

\(A \times B \neq B \times A\)   (i.e., \(A \times B\) and \(B \times A\) are not equal)

But, \(n(A \times B) = n(B \times A)\)

(Note: '\(n\)' denotes number of elements of a set)

6) Empty Set Condition

\(A \times B = \phi\)  \(\Rightarrow\)  is \(A = \phi\) (or) \(B = \phi\)

7) Number of Elements Formula

If \(n(A) = p\) and \(n(B) = q\), then \(n(A \times B) = pq\).

8) Distributive Properties

• \(A \times (B \cup C) = (A \times B) \cup (A \times C)\)
• \(A \times (B \cap C) = (A \times B) \cap (A \times C)\)