Question

Show that, given only the less-than operator ( \(<\) ) and the boolean operators and \((\& \&),\) or \((\|),\) and not \((!),\) it is possible to implement all of the other comparators: \(>,<=,>=,==\)

Short answer

> is !(a < b),<= is (a < b) || (a == b),>= is !(a < b), and == is !(a < b) && !(b < a).

Step-by-step solution

- Implement Greater Than (>)

To show that it's possible to implement 'greater than' using the 'less-than' operator and boolean operators, recognize that if 'a' is not less than 'b' and 'a' is not equal to 'b', then 'a' must be greater than 'b'. Formally, this can be written as: a > b = !(a < b)

- Implement Less Than or Equal To (

The 'less than or equal to' comparison can be derived from the 'less than' operator by including equality. So, 'a' is less than or equal to 'b' if 'a' is less than 'b' or 'a' is equal to 'b'. Formally,a <= b = (a < b) || (a == b)

- Implement Greater Than or Equal To (>=)

The 'greater than or equal to' is the inverse of 'less than', which can be defined as 'not less than'. Formally,a >= b = !(a < b)

- Implement Equals (==)

Two values 'a' and 'b' are equal if neither 'a' is less than 'b' nor 'b' is less than 'a'. Formally,a == b = !(a < b) && !(b < a)

Key concepts

less-than operator

In C++, comparison operators are crucial for making comparisons between values. One such operator is the less-than operator ( < ). This operator checks if the value on the left side is smaller than the value on the right side. Understanding this operator is the first step to mastering inequalities and logic in programming.
The syntax is straightforward: suppose we have two variables, `a` and `b`. When we write `a < b`, it will return `true` if `a` is indeed less than `b`.
Here's a simple application:

• int a = 5;
• int b = 10;
• bool result = a < b; // result will be true because 5 is less than 10

By understanding the less-than operator, you can start building more complex logical structures like inequalities and decision-making statements in your code.

boolean operators

Boolean operators in C++ are used to perform logical operations. They include 'and' (&&), 'or' (||), and 'not' (!). These operators are used to combine multiple conditions or to inverse a condition.
- **AND (&&):** Returns `true` if both operands are true.
- **OR (||):** Returns `true` if at least one of the operands is true.
- **NOT (!):** Inverts the value of the operand.
Let's see how these operators work through some examples:

• bool a = true;
• bool b = false;
• bool result1 = a && b; // result1 will be false because only one operand is true
• bool result2 = a || b; // result2 will be true because at least one operand is true
• bool result3 = !a; // result3 will be false because 'a' is true and NOT operator inverts it

By combining the less-than operator with these Boolean operators, you can derive more advanced comparison operators, as demonstrated in the exercise’s solution.

inequalities in programming

Inequalities are fundamental in programming. They help determine relationships between variables. Besides the basic less-than ( < ) operator, we have several other inequalities: 'greater-than' (>), 'less-than or equal to' ( <= ), and 'greater-than or equal to' ( >= ). These can all be derived using the less-than operator and Boolean operators, enhancing your programming logic.
For instance,

• To implement 'greater-than', we use inversion: a > b = !(a < b)
• The 'less-than or equal to' is expressed as: a <= b = (a < b) || (a == b)
• The 'greater-than or equal to' is derived by negating 'less-than': a >= b = !(a < b)
• Equality is a combination of two less-than comparisons: a == b = !(a < b) && !(b < a)

Understanding these concepts allows you to implement more complex logic in your code efficiently. Always practice these operators to get a better grasp of how they interact and work together to create meaningful conditions.