a) Let M be a matching in a graph G such that G has no M-augmenting path. Show that M is a maximum matching. Let G be a bipartite graph with bipartition (X,Y). For SCX, let Ej be the set of edges in G incident on some vertex in S, and let E2 be the set of edges in G incident with some vertex in N(S). Is it true in general that E¡ C E2? Why? Prove that if G is a graph with no isolated vertices, then a'(G)+B'(G) = n(G) b) d) Which of the following is true ? Justify. i) Every tree has a perfect matching. ii) Every tree has at most one perfact matching.

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Magnitude

A vector is an element with both direction and magnitude. A vector is represented using an arrow, where the arrow represents the direction of the vector and its length is represented using the magnitude. The set of vectors in space is known as vector space. The vectors can be added, subtracted, multiplied, and divided in a vector space. The image below shows a vector AB.