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3. Use the definition of convex sets to answer the following: (a) Show that if the sets S and T are convex, then SnT is convex. (b) Show that the intersection of any number of convex sets is convex. (c) A hyperplane in Rd is a set of points of the form {r: ax=b} where a R and b E R. Show that hyperplanes are convex. Hint: If you're having trouble seeing why this is true in general, try the problem with a simple concrete example in 2-dimensions.

3. Use the definition of convex sets to answer the following: (a) Show that if the sets S and T are convex, then SnT is convex. (b) Show that the intersection of any number of convex sets is convex. (c) A hyperplane in Rd is a set of points of the form {r: ax=b} where a R and b E R. Show that hyperplanes are convex. Hint: If you're having trouble seeing why this is true in general, try the problem with a simple concrete example in 2-dimensions.

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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3. Use the definition of convex sets to answer the following: (a) Show that if the sets S and T are convex, then SnT is convex. (b) Show that the intersection of any number of convex sets is convex. (c) A hyperplane in Rd is a set of points of the form {r: ax=b} where a € Rd and bER. Show that hyperplanes are convex. Hint: If you're having trouble seeing why this is true in general, try the problem with a simple concrete example in 2-dimensions. (d) A halfspace R is a set of points of the form {r: a'r <b} where a € Rd and bER. Show that halfspaces are convex. (e) Using (a) and (d), show that {r:c<a¹a<b} is convex when c < b. (f) Using (b) and (e), show that the d-dimensional cube {a: 0≤x≤1 for i {1,..., d}} is convex
Transcribed Image Text:3. Use the definition of convex sets to answer the following: (a) Show that if the sets S and T are convex, then SnT is convex. (b) Show that the intersection of any number of convex sets is convex. (c) A hyperplane in Rd is a set of points of the form {r: ax=b} where a € Rd and bER. Show that hyperplanes are convex. Hint: If you're having trouble seeing why this is true in general, try the problem with a simple concrete example in 2-dimensions. (d) A halfspace R is a set of points of the form {r: a'r <b} where a € Rd and bER. Show that halfspaces are convex. (e) Using (a) and (d), show that {r:c<a¹a<b} is convex when c < b. (f) Using (b) and (e), show that the d-dimensional cube {a: 0≤x≤1 for i {1,..., d}} is convex
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