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// Implement the IMyContraVarGenIF interface class MyClass<T> : IMyContraVarGenIF<T> { public void Show(T x) { ConsoleWriteLine(x); } }

Here, Show( ) simply displays the string representation of x (as obtained by WriteLine( ) s implicit call to ToString( )) Next, a class hierarchy is declared:

The number systems that we use in calculus are the natural numbers, the integers, the rational numbers, and the real numbers Let us describe each of these:

// Create a simple class hierarchy class Alpha { public override string ToString() { return "This is an Alpha object"; }

18:

// }

ing to sabotage me or he was incompetent, but I knew I had to do something My instinct was to attack read my rst draft The Empathy Index saved me from making the situation worse by forcing me to keep my focus on what was important Rather than berate Jim, a pointless exercise, it made more sense for me to maintain a calm tone As you ll see in the upcoming chapters, there are lots of approaches to revising first drafts We ve seen that there are several ways to write well You ll see that there are several ways to revise well, too Here is Paula s revision:

class Beta : Alpha { public override string ToString() { return "This is a Beta object"; } // }

The natural numbers are the system of positive counting numbers 1, 2, 3, We denote the set of all natural numbers by N The integers are the positive and negative whole numbers: , 3, 2, 1, 0, 1, 2, 3, We denote the set of all integers by Z The rational numbers are quotients of integers Any number of the form p/q, with p, q Z and q = 0, is a rational number We say that p/q and r/s represent the same rational number precisely when ps = qr Of course, you know that in displayed mathematics we write fractions in this way:

Notice that these versions of Alpha and Beta differ from the previous example for the sake of illustration Also notice that ToString( ) is overridden to return the type of object Given the foregoing, the following sequence is legal:

// Create an IMyContraVarGenIF<Alpha> reference to a // MyClass<Alpha> object // This is legal with or without contravariance IMyContraVarGenIF<Alpha> AlphaRef = new MyClass<Alpha>(); // Create an IMyContraVarGenIF<beta> reference to a // MyClass<Beta> object // This is legal with or without contravariance IMyContraVarGenIF<Beta> BetaRef = new MyClass<Beta>(); // Create an IMyContraVarGenIF<beta> reference to // a MyClass<Alpha> object // *** This is legal because of contravariance *** IMyContraVarGenIF<Beta> BetaRef2 = new MyClass<Alpha>(); // This call is legal with or without contravariance BetaRefShow(new Beta()); // Assign AlphaRef to BetaRef // *** This is legal because of contravariance *** BetaRef = AlphaRef; BetaRefShow(new Beta());

First, notice that two IMyContraVarGenIF reference variables are created and are assigned references to MyClass objects whose type parameters match that of the interface references The first uses Alpha The second uses Beta These declarations do not require contravariance and are legal in all cases Next, an IMyContraVarGenIF<Beta> reference is created, but it is assigned a reference to a MyClass<Alpha> object This is legal only because T is contravariant As you would expect, the next line, which calls BetaRefShow( ) with a Beta argument, is legal because T in MyClass<Beta> is Beta, and the argument to Show( ) is Beta The next line assigns AlphaRef to BetaRef This is legal only because of contravariance In this case, BetaRef is of type MyClass<Beta>, but AlphaRef is of type MyClass<Alpha> Because Alpha is a base class of Beta, contravariance makes this conversion legal To prove to yourself that contravariance is required in the program, try removing in from the declaration of T in IMyContraVarGenIF Then attempt to recompile the program As you will see, errors will result

7 1 2 + = 2 3 6

Part I:

For your convenience, all the pieces are assembled into the following program: