Day 02-Discrete mathematics for gate – Concept of Cartesian product and Introduction of Relation.

Day 02-Discrete mathematics for gate – Concept of Cartesian product and Introduction of Relation.

Here’s a clear explanation of Day 02: Discrete Mathematics for GATE focusing on the Concept of Cartesian Product and an Introduction to Relations, tailored for GATE CSE/IT aspirants. Bilingual format (English + Hindi) for better understanding.

Day 02 – Discrete Mathematics for GATE

Topics Covered:

• Cartesian Product (कार्तीय गुणनफल)

• Introduction to Relations (संबंध का परिचय)

1. Cartesian Product – Concept

English Definition:

The Cartesian product of two sets A and B, written as A × B, is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

Hindi Meaning:

दो सेट A और B का Cartesian Product, यानि A × B, उन सभी ordered pairs (a, b) का सेट है जहाँ a सेट A से है और b सेट B से है।

Formula:

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

Example:

Let:
A = {1, 2}
B = {x, y}

Then:

$$A\times B=\{(1,x),(1,y),(2,x),(2,y)\}$$ $$|A\times B|=|A|\times |B|=2\times 2=4$$

Important Notes:

• A × B ≠ B × A (Order matters)

• If A or B is empty, A × B is also empty.

2. Introduction to Relations

English Definition:

A relation R from set A to set B is a subset of the Cartesian product A × B.

That is,

$$R\subseteq A\times B$$

Hindi Explanation:

Relation (संबंध) एक ऐसा सेट होता है जो A × B के कुछ selected ordered pairs को चुनता है। यह एक subset होता है।

Example of Relation:

Let:
A = {1, 2, 3}
Define relation R = “a is less than b”
Then B = A, and:

$$R=\{(1,2),(1,3),(2,3)\}$$

यह relation R A से B पर define किया गया है, और यह A × B का एक subset है।

Types of Relations (for upcoming sessions):

GATE Point of View:

• Cartesian product is frequently used to define Relations, Functions, Graphs, and Matrices.

• You may get 1 mark MCQs asking for number of elements in A × B or identifying type of relation.

• Always practice Venn diagrams and arrow diagrams to visualize relations.

Practice Questions:

1. If A = {1, 2}, B = {a, b, c}, find A × B and its cardinality.

2. Let A = {1, 2, 3}, define R = {(a, b) | a divides b}. Write R.

3. How many relations are possible from A = {1, 2} to B = {3, 4, 5}?

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Day 02-Discrete mathematics for gate – Concept of Cartesian product and Introduction of Relation.

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