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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.

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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:


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×B={(a,b)∣a∈A, b∈B}A × B = \{ (a, b) \mid a \in A,\ b \in B \}


Example:

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

Then:

A×B={(1,x),(1,y),(2,x),(2,y)}A × B = \{(1, x), (1, y), (2, x), (2, y)\} ∣A×B∣=∣A∣×∣B∣=2×2=4|A × B| = |A| × |B| = 2 × 2 = 4


Important Notes:


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⊆A×BR \subseteq A × 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)}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):

Type Example Condition
Reflexive (a, a) ∈ R for all a ∈ A
Symmetric (a, b) ∈ R ⇒ (b, a) ∈ R
Transitive (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R
Antisymmetric (a, b) ∈ R & (b, a) ∈ R ⇒ a = b

GATE Point of View:


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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