Question

if A={1,2,3,4} and x,y€A,form the set of all ordered pairs (x,y) such that x is a divisor of y​

Answer 1

SOLUTION

GIVEN

$\sf{A = \{ 1,2,3,4 \} \: \: and \: \: x, y \in \: A}$

TO DETERMINE

The set of all ordered pairs (x,y) such that x is a divisor of y

CONCEPT TO BE IMPLEMENTED

CARTESIAN PRODUCT

For any two sets A and B, the Cartesian product of A and B is denoted by A × B and defined as

$\sf{A \times B = \{(x, y) : x \in A \: , y \in \: B \}}$

EVALUATION

Here

$\sf{A = \{ 1,2,3,4 \} \: \: and \: \: x, y \in \: A}$

Now we are proceeding to find the set of all ordered pairs (x,y) such that x is a divisor of y

Let S be the required Set

$\sf{Since \: \: 1 \: divides \: 1 \: , \: so \: \: (1,1) \in \: S}$

$\sf{Since \: \: 1 \: divides \: 2 \: , \: so \: \: (1,2) \in \: S}$

$\sf{Since \: \: 1 \: divides \: 3 \: , \: so \: \: (1,3) \in \: S}$

$\sf{Since \: \: 1 \: divides \: 4\: , \: so \: \: (1,4) \in \: S}$

$\sf{Since \: \: 2 \: divides \: 2 \: , \: so \: \: (2,2) \in \: S}$

$\sf{Since \: \: 2 \: divides \: 4 \: , \: so \: \: (2,4) \in \: S}$

$\sf{Since \: \: 3 \: divides \: 3 \: , \: so \: \: (3,3) \in \: S}$

$\sf{Since \: \: 4 \: divides \: 4 \: , \: so \: \: (4,4) \in \: S}$

Hence

$\sf{S = \{ (1,1),),(1,2),(1,3),(1,4),(2,2),(2,4),(3,3),(4,4) \}}$

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