Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Please do Parts D,E,F please.
Thanks
![2. Determine with proof if each of the following sets below is a convex set?
1
(a) {x €": [x] ≤ r}, r> 0 is a real number.
(b) {x €¹: Ax = b}, A is an m × n matrix and b €m.
(c) {x ¤”: x ≥ 0}, where x ≥ 0 means that every component of x is
nonnegative.
(d) A linear variety.
(e) If S₁ and S₂ are convex sets, then
S₁ + S₂ = {x : x = V₁ + V2, V₁ € S₁, V₂ € S₂}
is also a convex set.
(f) The intersection of any collection of convex sets is convex.
To prove that a set is convex, we have to prove that for any x, y = N,
and any 0 ≤ a ≤ 1, the point ax + (1 − a)y is also in N.](https://content.bartleby.com/qna-images/question/7b12921e-4f95-442d-998d-c25c97115157/9e2b86cd-d3fe-484e-8178-9703af836830/apt9hu_processed.png)
Transcribed Image Text:2. Determine with proof if each of the following sets below is a convex set?
1
(a) {x €": [x] ≤ r}, r> 0 is a real number.
(b) {x €¹: Ax = b}, A is an m × n matrix and b €m.
(c) {x ¤”: x ≥ 0}, where x ≥ 0 means that every component of x is
nonnegative.
(d) A linear variety.
(e) If S₁ and S₂ are convex sets, then
S₁ + S₂ = {x : x = V₁ + V2, V₁ € S₁, V₂ € S₂}
is also a convex set.
(f) The intersection of any collection of convex sets is convex.
To prove that a set is convex, we have to prove that for any x, y = N,
and any 0 ≤ a ≤ 1, the point ax + (1 − a)y is also in N.
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