8B. Mental Model of Boolean Expressions

Translating from English to C++.

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Be careful when translating from English to C++. English allows you to take liberties that C++ does not. Common mathematical notation also allows you to write things that C++ does not allow.

x < y < z

You cannot write combinations of comparisons this way. Write x < y && y < z.

x is 1 or 2

Do not try to write this as x == 1 || 2. Each operand of || is treated as a boolean expression. Expression x == 1 || 2 is the same as (x == 1) || (2), which treats 2 as a boolean value (which is considered to be true). To ask if x is 1 or 2, say x ==1 || x == 2.

x is even

In English, is sometimes means "is equal to", and sometimes means "has the property". For example, "Jumbo is an elephant" does not indicate that "Jumbo" and "an elephant" are the same thing, but that Jumbo has the property of being an elephant. In C++, some functions, called predicates, yield boolean values. Predicates are used to check for properties. For example, suppose that even is a predicate where even(x) yields true if x is even and false if x is odd. To use a predicate, treate it like any other function. If you want to ask if x is even, write even(x). Do not write x == even. If you have a function isPrime(x) that returns true if x is prime, then test isPrime(n) works to check whether n is prime. Do not write x == isPrime.

isPrime(n) is true
isPrime(n) is false

To ask if n is prime, just write isPrime(n). Do not write isPrime(n) == true. To test whether n is not prime, write !isPrime(n), not isPrime(n) == false.

Using boolean values as values

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Shortly, we will see if-statements, which are similar to if-statements in Java. Some students get the idea that a boolean expression can only occur in a test, such as in an if-statement. But that is not true. If x is 2, then expression x > 0 has value true. Instead of

  if(x > 0)
  {
    return true;
  }
  else
  {
    return false;
  }

  return x > 0;

  if(x > y)
  {
    v = false;
  }
  else
  {
    v = true;
  }

  v = !(x > y);

  v = x <= y;

Exercises

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1. Give an expression that is equivalent to !(x == y), but that does not use the negation operator, !. Answer

2. Give an expression that is equivalent to !(x > y), but that does not use the negation operator, !. Answer

3. Give an expression that is true if x, y and z are all equal to one another. Answer

4. Give an expression that is equivalent to isPrime(n) == false, but that does not use any comparison operator. Answer