The Negation of a Statement

After creating statements, we wish to form more complex statements through the use of words such as "not", "and", "or", and "implies".  The aforementioned four words are often referred to as logical connectives.  In order to interpret the behavior of each of these logical connectives we create a truth table.  A truth table is a table showing what the resulting truth value of a complex statement is for all the possible truth values for the simple statements.

The Negation

The negation of a statement P is the statement

not P.  

In order to wrap our heads around this new concept, we shall look at a few examples.

Examples of Negations

  1.   Consider the statement

P:  The Eiffel tower is in Budapest.

            The negation of this statement can be described in a couple of ways.  The first will be by inserting the word "not" in the appropriate place.

not P:  The Eiffel tower is not in Budapest.

            For our second case of negating a statement, we can complete this task by stating "It is not the case that P."  We provide such an example below.

not P:  It is not the case that the Eiffel tower is in Budapest.

            We often avoid using the second method since it is much more cumbersome than the first method.

      2.   Let's begin with the statement

P:  5 < 9

           The negation for this statement is given by 

not P:  5 is not less than 9

           The beauty of this statement is that it can be stated in a multitude of ways, which we shall provide an example of here. 

not P:  5 is greater than or equal to 9

While we formed the negation of these statements, we neglected to look at the truth value of the original statement and its negation.  We shall revisit each example here.

Examples of Negations (Revisited)

  1. Notice that the Eiffel tower is located in Paris, France, instead of the claimed location of Budapest.  Therefore, the statement P is false.  However, since the Eiffel tower is not in Budapest, the negation of the statement P (also referred to as "not P") is true.
  2. For this problem, we find that the statement P is true since 5 is less than 9.  However, not P is false since the statement "5 is not less than 9" would wreak havoc throughout all of mathematics.

The previous examples help illustrate that for the truth value of a statement P the negation of the statement P would take on the opposite truth value.  These ideas are the basis for creating a truth table where T shall stand for true and F stands for false.

P not P
T F

F T

If we are ever in doubt about the truth value of the negation of a statement, we can consult the above truth table to find our answer.  This will come in handy as we shall look for truth values when looking at other logical connectives.

Learning Activities

In order to further comprehend the material, please attempt the following exercises.

    1. Find the negation of the following statements.
    2. Determine the truth value of the negation of the following statements.