The given question is : Consider the sets defined by the real solutions of the inequalities A

hicaderasu1c8q

hicaderasu1c8q

Answered question

2022-06-03

The given question is :
Consider the sets defined by the real solutions of the inequalities
A = { ( x , y ) : x 2 + y 4 1 } B = { ( x , y ) : x 4 + y 6 1 }
A. B A
B. A B
C. Each of the sets A B, B A and A B is non-empty
D. None of the above
how to solve this without using any such plotter?

Answer & Explanation

Siffkiffanewsler6n

Siffkiffanewsler6n

Beginner2022-06-04Added 7 answers

Note that if | x | > 1 or | y | > 1, then ( x , y ) does not belong to A or B. Suppose ( x , y ) A, so that x 2 + y 4 1. Then we have that
x 4 + y 6 = x 2 × x 2 + y 2 × y 4 1 2 × x 2 + 1 2 × y 4 = x 2 + y 4 1.
It follows that ( x , y ) B and thus A B.
oplodnjomqunfs

oplodnjomqunfs

Beginner2022-06-05Added 2 answers

All points ( x , y ) in A or B must satisfy | x | 1 and | y | 1. Under these conditions, you have that x 4 x 2 and y 6 y 4 . So,

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