Question

Consider the following subsets of R2:
C1 ={(x,y)∈R2 :x+y≤3,x≥0,y≥0}

C2 ={(x,y)∈R2 :4x+y≤4,x≥0,y≥0}

Algebra Consider the following subsets of R2. Draw a sketch of the intersection CinC2 and the union C1UC2. State whether each set is convex or not. If the set is not convex, give an example of a line segment for which the definition of convexity breaks down

Answer 1

A graph of the lines x+y = 3 (AP)and 4x+y = 4 (BQ)) is attached. The set C₁ is the part of the 1^st quadrant on and below the red line and the set C₁ is the part of the 1^st quadrant on and below the blue line.

The set C₁∩ C₂ is represented by the area A O Q M, where A is the point where line x+y = 3 meets the Y-axis, O is the origin , Q is the point where the line 4x+y = 4 meets the X-axis and M is the point of intersection of the 2 lines. The area includes the segment AM of the line AP and the segment MQ of the line BQ.

Now, if we select any 2 points X,Y in the set C₁∩ C₂, then the line XY is entirely within the set C₁∩ C₂. Hence the set C₁∩ C₂ is a convex set.

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