Discrete mathematics question. Please provide a reasonable explanation with your answer.

In this problem, you will show that given n red points and n blue points in the plane such that no three points lie on a common line, it is possible to draw line segments between red-blue pairs so that all the pairs are matched and none of the line segments intersect. Assume that there are n red and n blue points fixed in the plane. A matching M is a collection of n line segments connecting distinct red-blue pairs. The total length of a matching M is the sum of the lengths of the line segments in M. Say that a matching M is minimal if there is no matching with a smaller total length Let IsMinimal(M) be the predicate that is true precisely when M is a minimal matching. Let HasCrossing(M) be the predicate that is true precisely when there are two line segments in M that cross each other Give an argument in English explaining why there must be at least one matching M so that IsMinimal(M) is true, i.e. EMIsMinimal(M)) Give an argument in English explaining why VM (HasCrossing (M) -Minimal(M) Now use the two results above to give a proof of the statement: 3M-HasCrossing(M) In this problem, you will show that given n red points and n blue points in the plane such that no three points lie on a common line, it is possible to draw line segments between red-blue pairs so that all the pairs are matched and none of the line segments intersect. Assume that there are n red and n blue points fixed in the plane. A matching M is a collection of n line segments connecting distinct red-blue pairs. The total length of a matching M is the sum of the lengths of the line segments in M. Say that a matching M is minimal if there is no matching with a smaller total length Let IsMinimal(M) be the predicate that is true precisely when M is a minimal matching. Let HasCrossing(M) be the predicate that is true precisely when there are two line segments in M that cross each other Give an argument in English explaining why there must be at least one matching M so that IsMinimal(M) is true, i.e. EMIsMinimal(M)) Give an argument in English explaining why VM (HasCrossing (M) -Minimal(M) Now use the two results above to give a proof of the statement: 3M-HasCrossing(M)