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1. In this exercise you will prove that disc {(x, y) : x+y < 1} is an open subset of R, and then that every open disc in the plane is an open set. {(x, y) : x2 + y? < 1}. Va? + b2. Let R, ) be the open rectangle with vertices at the points (i) Let (a, b) be any point in the disc D Put r = (a +, 6+3"). Verify that Ra.b) C D. (ii) Using (i) show that D= U R(a.b) (a,b)ED (iii) Deduce from (ii) that D is an open set in R4. (iv) Show that every disc {(x, y) : (r- a)2 + (y- b)2 < c, a, b, c e R} is open in R2.

1. In this exercise you will prove that disc {(x, y) : x+y < 1} is an open subset of R, and then that every open disc in the plane is an open set. {(x, y) : x2 + y? < 1}. Va? + b2. Let R, ) be the open rectangle with vertices at the points (i) Let (a, b) be any point in the disc D Put r = (a +, 6+3"). Verify that Ra.b) C D. (ii) Using (i) show that D= U R(a.b) (a,b)ED (iii) Deduce from (ii) that D is an open set in R4. (iv) Show that every disc {(x, y) : (r- a)2 + (y- b)2 < c, a, b, c e R} is open in R2.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. In this exercise you will prove that disc {(x, y) : x+y <1} is an open subset of R, and then that every open disc in the plane is an open set. x2 + y? < 1}. Put Va2 + b2. Let R, be the open rectangle with vertices at the points (1) Let (a, b) be any point in the disc D = {{x, y) : r = (a +, 6+). Verify that Rab) C D. (ii) Using (i) show that D = U R(a.b)- {a,b)ED (iii) Deduce from (ii) that D is an open set in R4. (iv) Show that every disc {(x, y) : (x- a)2 + (y-b)2 < c², a, b, c e R} is open in R2.
Transcribed Image Text:1. In this exercise you will prove that disc {(x, y) : x+y <1} is an open subset of R, and then that every open disc in the plane is an open set. x2 + y? < 1}. Put Va2 + b2. Let R, be the open rectangle with vertices at the points (1) Let (a, b) be any point in the disc D = {{x, y) : r = (a +, 6+). Verify that Rab) C D. (ii) Using (i) show that D = U R(a.b)- {a,b)ED (iii) Deduce from (ii) that D is an open set in R4. (iv) Show that every disc {(x, y) : (x- a)2 + (y-b)2 < c², a, b, c e R} is open in R2.
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