If I had a class Airplane and a class Wing, if there was a composition relationship between the two, does Airplane have a member variable of type Wing in the class diagram, shown in the Airplane box?
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If I had a class Airplane and a class Wing, if there was a composition relationship between the two, does Airplane have a member variable of type Wing in the class diagram, shown in the Airplane box?
No. Compositions and aggregations are kinds of associations and are shown like associations, i.e., with lines between classes (with solid and hollow diamonds, respectively, on the containing side). As a general rule, if you have an attribute whose type is a class, your model is wrong.
Implementation is a completely separate matter from analysis/design. You may implement associations in a variety of ways, including using member variables e.g. in C++.