Bounded and convex sets?

Question

are the following:
-bounded or unbounded??
-convex sets or not convex sets??

a)
[ ( x , y ) І x + y ≤ 1, x - y ≤ 2 , x ≥ 0 ]

b)
[ ( x , y ) І ІxІ ≤ 1, Іx - 3І ≤ 2 ]

c)
[ ( x , y ) І x² + y² ≤ 1 and x² + (y² + 1) ≤ 1 ]

d)
[ ( x , y ) І (x - 2)² + y² ≤ 1 or x - y ≤ 0 ]

cheers for your help!!

Answer 1

a) I assume you meant to include "and".

[ ( x , y ) І x + y ≤ 1 and x - y ≤ 2 and x ≥ 0 ]

This is bounded and convex.

b) I assume you meant to include "and".

[ ( x , y ) І ІxІ ≤ 1and Іx - 3І ≤ 2 ]

-1 ≤ x ≤ 1

-2 ≤ x - 3 ≤ 2
1 ≤ x ≤ 5

Solution set is the line x = 1.

This is unbounded. I don't think convex / not convex applies to a line.

c)
[ ( x , y ) І x² + y² ≤ 1 and x² + (y² + 1) ≤ 1 ]

These are two partially overlapping circles.

This is bounded and convex.

d)
[ ( x , y ) І (x - 2)² + y² ≤ 1 or x - y ≤ 0 ]

The second condition contains the first so this is just the half-plane

x - y ≤ 0

This is unbounded.

Answer 2

The above answer proves plenty greater suitable than what you like. Any functionality on the airplane with non-supply up by-product is lipschitz on bounded contraptions. Follows from the definition and integration on segments. that's it.

Answer 3

I'm sorry...does this question involve Rosalind Russell???

...whoops...my mistake...