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In some cases, A union B is convex, but in general this may not be true. Consider two sets A, B (subsets of Rn) such that A intersect B is the null set. Now choose a point x in A, and y in B. If a set is to be convex, then all points on the line tx + (1-t)y (0 <= t <= 1) must lie in A union B.
However this is clearly not the case since A intersect B is the null set. Therefore A union B is not convex in general.
Example:
A = [0,1], B = [2,3]. Choosing any point in A and any point in B, we find that there will be points on the line tx + (1-t)y that are not in A or B.
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Is A union B a convex set?
yes
Intersection of two convex set is convex?
The proof of this theorem is by contradiction. Suppose for convex sets S and T there are elements a and b such that a and b both belong to S∩T, i.e., a belongs to S and T and b belongs to S and T and there is a point c on the straight line between a and b that does not belong to S∩T. This would mean that c does not belong to one of the sets S or T or both. For whichever set c does not belong to this is a contradiction of that set's convexity, contrary to assumption. Thus no such c and a and b can exist and hence S∩T is convex.
Give an example to show that if A union B and A intersection B be given then A and B may not be uniquely determined?
The set A union B can be decomposed into three disjoint sub sets A\ (A int B), B\(A int B), and (A int B). So in this case (A union B) and (A int B) are fixed but "moving" elelments from A\ (A int B) into B\(A int B) will not affect (A union B) and (A int B). You should be able to fill in the details now.
How will you write using the venn diagram the union of set a(1234) and the intersection of set b(10111213) and set c(1213)?
If B = {10111213} and C = {1213} then their intersection is the empty set, {}.The union of A with an empty set is set A.
What do you call a set of elements which belong to A or to B or to both?
The union of A and B.
