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One of the most common everyday objects that we use in our daily life is a computer. A computer is made up of complex circuitry and a lot of other algorithms that control how a computer operates and performs.
But at its core, a computer is nothing but a combination of Boolean logic and a series of zeros and ones. Hence, it is essential to have the skill to evaluate Boolean expressions and understand what is going to be the output of an expression given a set of inputs.
In order to achieve this, we have a tool called truth tables that helps us evaluate Boolean expressions.
What is a Truth Table?
Truth tables have been in use since late 1893 and while there is no clear answer as to who invented the truth tables, Ludwig Wittgenstein is credited with popularising the truth table and is also often credited with inventing the truth tables too.
In its essence, a truth table is nothing but a mathematical tool that is specifically applied in the domain of logic when the task at hand has connections with Boolean algebra or functions for propositional calculus.
A truth table is formed for a given logical expression when its output has to be evaluated given a certain combination of its input arguments. In terms of a function, a truth table can be expressed as the combination of all the functional values produced by a logical expression for a given set of combinations of functional arguments submitted as input.
Popular usage of the truth table is to prove that a given propositional expression is logically valid; that is, for all available input values, the expression produces a valid output. A truth table consists of rows and columns. There are n number of columns for n number of input variables and one column for the output variable. There are r number of rows for r possible combinations of the input variables. Every row in a truth table is distinct from other rows.
How to create a Truth Table?
There are two basic requirements for creating a truth table. The first requirement is knowing all the input variables and the second requirement is knowing the logical expression that needs to be evaluated.
Once we have all the requirements, we start building the truth table by creating the number of columns required by the input variables and then adding an extra column for the output variable, which is the expression that has to be evaluated. Let us try to understand how to build a truth table via an example.
A person wants to buy a laptop with either good RAM power or an SSD. Find all the cases using a truth table in which the requirement of the person can be met.
In this situation, let us assume that having a good RAM is denoted by R and an SSD by S. According to the requirement of the person, the laptop must have either R or S. Modelling this in the truth table we get:
R | S | R or S |
F | F | F |
T | F | T |
F | T | T |
T | T | T |
In this truth table, T denotes truth and F denotes false. In the first scenario, we can see that both R and S are false, and hence the laptop does not have a good Ram or an SSD; therefore, it does not meet the requirement of the person.
In the second and the third case, either one of the variables is true. That is, the laptop either has a good RAM or has a good SSD for these conditions to meet the requirement of the person and hence the output is true.
In the fourth case, the laptop has both a good RAM and a good SSD, and obviously, this also meets the requirement of the person, and hence the output is true.
Sample Questions of Truth Table
Let us try to understand a few easy ways to solve truth table problems with the help of solved examples of truth table problems.
Given the following expression, create the truth table for the output of the expression
(A∧B)∨C
Here, ∧ stands for logical AND operation and ∨ stands for logical OR operation
The first step to solving truth table questions is to understand the input variables that are given in the expression. In this logical expression, there are 3 input variables A, B and C.
Also, there are two operators that are used in this expression, logical AND and logical OR. Therefore the structure of the truth table will be as follows.
A | B | C | A∧B | (A∧B)∨C |
0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 1 |
0 | 1 | 0 | 0 | 0 |
0 | 1 | 1 | 0 | 1 |
1 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 0 | 1 |
1 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 1 | 1 |
In this table, we can observe that we have broken down a complex expression with more than one operator into two expressions in order to solve the truth table.
In this manner, if a logical expression has more than one operator, then it can be broken down into smaller expressions which can then be evaluated together in a truth table.
Let us see another example for creating a truth table for a logical statement. Create the truth table for the statement “If there is sun it is day”.
Let S denote the possibility of the sun being present and D denote the possibility of a day. The statement can be modelled in the form of a Boolean expression as;
S∧D
The truth table for this is:
S | D | S∧D |
0 | 0 | 0 |
0 | 1 | 0 |
1 | 0 | 0 |
1 | 1 | 1 |
In the first scenario, both the input variables are false, and hence the output variable is false. This is a correct output because if there is no sun, there is no day.
In the second combination, the logical statement that can be formed is that it is a day without the sun. The output variable is false because such a scenario is impossible.
The 3rd combination can be modelled as a logical statement as there is a sun, but it is not during the day. Again the output is false because such a scenario cannot take place. The fourth combination can be modelled as follows: there is a sun and it is during the day. The output is true because this scenario is true.
Conclusion
Logical expressions form an important part of both mathematics and computers. In order to understand logical expressions and how they behave, the truth table is a very powerful tool. Truth tables can give an exact visualisation of how the output of a logical expression will change depending on the combination of input that is provided to the logical expression.
