Question

Let A={(x,y):x,y€z,x²+y²≤16},then the number of reflexive relations on set A is...a)2^49 b)2^2401 c)2^2352 d)2^1176​

Answer 1

SOLUTION

TO DETERMINE

$\sf{ A = \{(x, y) : x, y \in \mathbb{Z} \: and \: {x}^{2} + {y}^{2} \leqslant 16 \}}$

The number of reflexive relations on set A is

$\sf{}a) \: \: \: {2}^{49}$

$\sf{}b) \: \: \: {2}^{2401}$

$\sf{}c) \: \: \: {2}^{2352}$

$\sf{}d) \: \: \: {2}^{1176}$

CONCEPT TO BE IMPLEMENTED

If a set A contains n elements then the total number of reflexive relations on set A are

$\sf{ = {2}^{ {n}^{2} - n} \: }$

EVALUATION

Here

$\sf{ A = \{(x, y) : x, y \in \mathbb{Z} \: and \: {x}^{2} + {y}^{2} \leqslant 16 \}}$

CASE : 1 x = - 4

$\sf{Then \: \: {y}^{2} \leqslant 16}$

$\implies\sf{ \:y = 0 }$

So the points of A are (-4,0)

So the number of points of A is 1

CASE : 2 x = - 3

$\sf{Then \: \: {y}^{2} \leqslant 7}$

So y = - 2 , - 1 , 0 , 1 , 2

So the number of points of A are 5

CASE : 3 x = - 2

$\sf{Then \: \: {y}^{2} \leqslant 12}$

So y = - 3, - 2 , - 1 , 0 , 1 , 2, 3

So the number of points of A are 7

CASE : 4 x = - 1

$\sf{Then \: \: {y}^{2} \leqslant 15}$

So y = - 3, - 2 , - 1 , 0 , 1 , 2, 3

So the number of points of A are 7

CASE : 5 x = 0

$\sf{Then \: \: {y}^{2} \leqslant 16}$

So y = - 4, - 3, - 2 , - 1 , 0 , 1 , 2, 3, 4

So the number of points of A are 9

CASE : 6 x = 1

$\sf{Then \: \: {y}^{2} \leqslant 15}$

So y = - 2, - 2 , - 1 , 0 , 1 , 2, 3

So the number of points of A are 7

CASE : 7 x = 2

$\sf{Then \: \: {y}^{2} \leqslant 12}$

So y = - 3, - 2 , - 1 , 0 , 1 , 2, 3

So the number of points of A are 7

CASE : 8 x = 3

$\sf{Then \: \: {y}^{2} \leqslant 7}$

So y = - 2 , - 1 , 0 , 1 , 2

So the number of points of A are 5

CASE : 9 x = 4

$\sf{Then \: \: {y}^{2} \leqslant 0}$

So y = 0

So the number of points of A is 1

Therefore the total number of points of A are 49

Hence the total number of reflexive relations on set A are

$\sf{ = {2}^{ {49}^{2} - 49} \: \: }$

$\sf { = {2}^{49(49 - 1)} \: }$

$\sf { = {2}^{(49 \times 48)} \: }$

$\sf { = {2}^{2352} \: }$

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