Truth tables and the rule of not

What you will be able to do

After this lesson you can

• write a truth table for a combined condition,
• use it to check that two conditions are the same,
• turn a not (A and B) into a simpler form with the rule of not, and
• write a condition for a rule with an exception, such as the leap year.

Why a truth table?

A combined condition can be hard to judge in the head, especially with not and brackets. A truth table is a way to be sure. It lists every possible combination of True and False for the parts, and the result for each.

With two parts, a and b, there are four combinations. With three parts, eight. Let the computer write the table.

The rule of not

Look at this sentence: "It is not true that I am tired and hungry". It means I am either not tired, or not hungry, or both. The word and has turned into or when the not went inside. This is called the rule of not (also known as De Morgan's law, after the mathematician who wrote it down).

not (A and B) is the same as (not A) or (not B)

not (A or B) is the same as (not A) and (not B)

Let the truth table prove it for the first rule.

Why is it useful? Sometimes a condition is written with a not in front of a bracket, and it is hard to read. The rule gives a simpler form. For example,

if not (age >= 18 and has_ticket): is the same as if age < 18 or not has_ticket:.

Notice that not (age >= 18) became age < 18. A not in front of a comparison can be absorbed into the comparison: not (a > b) is a <= b.

A rule with an exception: the leap year

The rule for a leap year is: a year is a leap year if it is divisible by 4. But a century year (divisible by 100) is a leap year only if it is also divisible by 400. So 2024 is a leap year, 1900 is not, and 2000 is.

(year % 4 == 0 and year % 100 != 0) or year % 400 == 0

• The first bracket: divisible by 4 and not a century year. This is every normal leap year, such as 2024.
• The second part: divisible by 400. This brings back the century years like 2000.
• The two are joined with or: either way is enough.

The brackets are important. A rule with an exception is typical: an and with a not for the exception, and an or that gives the exception to the exception.

The brackets matter: the and part is the usual rule, and or year % 400 == 0 is the exception for century years.

How to build a hard condition

1. Write the rule in plain words, in small sentences.
2. Find the parts, and write each as a comparison.
3. Join them with and, or and not, and put in brackets.
4. Write a test table with the edges, and check every row (the previous chapter).

Common mistakes

• Not changing and into or when moving a not inside a bracket.
• Forgetting the brackets around the and part.
• Testing only the common years. The test years are 1900, 2000, 2100, 2024, 2023.
• Believing a condition is right because the examples worked. Check every row.

Practice

Start with two quick questions. Each New numbers gives you a fresh one, so you can practise as long as you like.

Then the labs and the problems:

1. Warm-up. Open the lab A truth table, if you have not done it, and the lab A leap year checker.
2. Build. Open the problem Is it a leap year? and the problem Predict: and, or, not.
3. Stretch. Open the problem Simplify with the rule of not and write the simpler condition.

Remember these

• A truth table lists every combination and the result. Two parts give four rows.
• not (A and B) equals (not A) or (not B). not (A or B) equals (not A) and (not B).
• not (a > b) is a <= b.
• A rule with an exception: and with not, and an or for the exception to the exception.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. How many rows does a truth table of three parts have?

2. Simplify not (x > 5 and y > 5).

3. Is 2100 a leap year?