Let A be a set of points. Let B be the union of all segments of the form line segment PQ, where P and Q belong to A. Does it follow that B is convex? Why or why not?

Here's what I think:

Yes, it follows that B is convex because if B is contained in A and P, Q are members of A the line segment PQ is in B, then B has to belong to A and the line segment PQ lies in A making B convex.

Is this right? Can you help me if it isn't please?

My book defines that convex as for every two points P,Q of A the entire segment PQ lies in A.

Thanks for all of the help!