Observation: You have a movie ticket, there are pair of informations:

• Your row number is 3 
• Your seat number is 10

If we put this information inside a parentheses $(   )$, separated by comma $,$ like $(3, 10)$ then we can say a pair of informations.

Observation: You have a movie ticket, there are pair of informations:

• Your row number is 3 
• Your seat number is 10

Practically your seat is (3, 10)

If you go on $10^{th}$ row and look for $3^{rd}$ seat then you will be on someone else seat, so order matter.

Conclusion: A pair of information written inside the parentheses, separated by comma following the order (rule) is called Ordered Pair.

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Definition:

An ordered pair is a pair of numbers, objects, elements or data written in a specific order (rule), usually inside parentheses separated by comma. like: $(x, y)$.

The first element in an ordered pair is called ANTECEDENT.

The second element in an ordered pair is called CONSEQUENT.

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Equality of Ordered Pairs:

Two ordered pairs (a, b) and (c, d) are said to be equal if and only if 

• a = c (antecedent = antecedent)
• b = d (consequent = consequent)

For example: ($\sqrt25$, $\sqrt36$) and (5, 6) are any two ordered pairs.

$1^{st}$ ordered pair:

Antecedent = $\sqrt25$ = 5

Consequent = $\sqrt36$ = 6

$2^{nd}$ ordered pair:

Antecedet = 5

Consequent = 6

Conclusion: Antecedent of $1^{st}$ ordered pair is equal to antecedent of $2^{nd}$ ordered pair

like wise, Consequent of $1^{st}$ ordered pair is equal to consequent of $2^{nd}$ ordered pair. 

We can say,  ($\sqrt25$, $\sqrt36$) and (5, 6) are equal ordered pairs.

 

Cartesian Product

Activity

Question: If you have 2 shirts and 3 pants, how many different outfits can you make?

Possible Outfits:

• Shirt 1 + Pant A
• Shirt 2 + Pant A
• Shirt 1 + Pant B
• Shirt 2 + Pant B
• Shirt 1 + Pant C
• Shirt 2 + Pant C

Total outfits: 6

 

Observation

• We have two collections (well defined and distinct):
  \( A = \{S_1, S_2\} \)
  \( B = \{P_A, P_B, P_C\} \)
• We are pairing one item from the first group with one item from the second group.
• We get a collection of ordered pairs:
  \( (S_1, P_A),\ (S_2, P_A),\ (S_1, P_B),\ (S_2, P_B),\ (S_1, P_C),\ (S_2, P_C) \)
• All pairs are ordered pairs because the order matters (shirt first, pant second).

 

Conclusion

The collection of all ordered pairs formed by taking one element from each of two sets is known as the Cartesian Product.

 

Definition

Let \( A \) and \( B \) be any two non-empty sets. The Cartesian Product of \( A \) and \( B \), written as \( A \times B \) (read as "A cross B"), is the collection of all ordered pairs \( (a, b) \) such that \( a \in A \) and \( b \in B \).

Mathematically:

\[ A \times B = \{(a,b) \mid a \in A,\ b \in B\} \]

 

Important Note: Order matters in ordered pairs.

\( (S_1, P_A) \neq (P_A, S_1) \)

Therefore, \( A \times B \neq B \times A \) (in general).

 

Cardinality of Cartesian Product

If \( n(A) = m \) and \( n(B) = n \), then

\[ n(A \times B) = n(A) \times n(B) = m \times n \]

In our example: \( n(A) = 2 \), \( n(B) = 3 \), so \( n(A \times B) = 6 \).

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Ways of Representing Cartesian Product

 

a) As Ordered Pairs:

Let \( A = \{1, 2, 3\} \), \( B = \{a, b\} \)

\( A \times B = \{(1,a),\ (1,b),\ (2,a),\ (2,b),\ (3,a),\ (3,b)\} \)

 

b) Tabular Form

c) Arrow Diagram / Mapping

d) Tree Diagram

e) Graphical Method (Coordinate Plane)