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A truth table is a mathematical table that is used to perform logical operations in mathematics. Boolean algebra and boolean functions are included. Based on the input values, it determines whether a compound statement is true or false. Each statement in a truth table is represented by the letters p,q, or r, and each statement in the truth table has its own column that lists all of the possible truth values. The output is the result of unary or binary operations performed on the input values. Binary operations include AND, OR, NOR, XOR, XNOR, and so on. Here, we will be learning the truth table in detail and discussing some important questions.
| Table of Content |
Key takeaways: Truth table, unary operations truth table, binary operations truth table, operational true tables, conditional operator.
Unary Operations Truth Table
Unary operation comprises a single input, which is either True or False. There are four unary operations for these inputs, which we are going to perform here. They are:
- Logical True (Only True)
- Logical False (Only False)
- Logical Identity
- Logical Negotiation
Logical True
The output is always true, despite any input value in logical true operation. Suppose P signifies the input values and Q signifies the output, then we can write the table as:
| P | Q→T |
| T | T |
| F | T |
Where T=True and F=False
Logical False
In comparison to the logical true, the output values for logical false are always false. It is also known as unary falsum. The table for this operation is given below.
| P | Q→F |
| T | F |
| F | F |
Where T=True and F=False
Logical Identity
The output value remains the same or equal to the input value in this operation. Let us find out with the help of the table.
| P | Q→P |
| T | T |
| F | F |
Logical Identity
We get the opposite value of the input value, as an output when we perform the logical negotiation operation on a single logical value or propositional value. Let us see the truth-table for this:
| P | Q→~P |
| T | F |
| F | T |
Read more: Set Operations
Binary Operations Truth Table
Two variables are used for input values in binary operations. The output result is dependent on the operation performed on the input or proposition values, and the value can be true or false. Some of the most important binary operations are:
- AND
- OR
- NAND
- NOR
- XOR
- Biconditional
- Conditional, also known as "if-then"
Now, for each binary operation, we will build the consolidated truth table using the input values X and Y.
| X | Y | AND | OR | NOR | NAND | XOR | Conditional | Bi-Conditional |
| T | T | T | T | F | F | F | T | T |
| T | F | F | T | F | T | T | F | F |
| F | T | F | T | F | T | T | T | F |
| F | F | F | F | T | T | F | T | T |
In the preceding table, T denotes true and F denotes false.
Truth Tables for NOR and OR
If any two input values are true, the operation OR statement represents that. The output value is always true. The symbol represents it ().
In contrast to the OR operation, the NOR operation produces output values. It implies that a statement that is true for OR is false for NOR and is denoted by (~∨).
Read more:
Operational True Tables NAND and AND
According to the above and operational true table, the output is true only if both input values are true; otherwise, the output is false. In the AND operational table, the AND operator is represented by the symbol (∧).
Truth Table for the XOR Operation
The table specifies that the input values must be either exactly true or exactly false. XOR is represented by the symbol (?).
Conditional Operator
The conditional operator, also known as the 'if-then' operator, returns True for all input values except when True implies False. It is denoted by the symbol ". This operation has the same logical meaning as the P Q operation. Let us demonstrate;
| P | Q | ~P | ~P ∨ Q |
| T | T | F | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |
You can compare the values of P⇒Q and ~P ∨ Q. Both are equivalent.
Logical equality is another name for bi-conditional. If both the values of P and Q are either True or False, the result is True; otherwise, the result is False.
Things to Remember
- In mathematics, a truth table is used to perform logical operations. These operations are referred to as boolean algebra or boolean functions. It is primarily used to determine whether a propositional expression is true or false based on the input values. The basis for this is boolean algebra.
- It is made up of columns for one or more input values, such as P and Q, and one column for the output results. The result of the unary or binary operation performed on the given input values is the output. Binary operations include AND, OR, NOR, XOR, XNOR, and so on. With their respective truth-tables.
- Unary functions have a single input that can be either True or False. There are four unary operations for these inputs that we will perform here. They are as follows: True Logical (Only True), False logically (Only False), Logical Identification, Logical Negotiation.
Sample Questions
Ques: Take a look at the following contingent statement.
y ∧ z∧ ¬x
What is the truth table for the preceding statement? (4 marks)
Ans:
| X | Y | Z | y ∧ z∧ ¬x |
|---|---|---|---|
| T | T | T | F |
| T | T | F | F |
| T | F | T | F |
| T | F | F | F |
| F | T | T | T |
| F | T | F | F |
| F | F | T | F |
| F | F | F | F |
Ques: Take a look at the following contingent statement.
x ∨ ¬ y ∨ ¬ z
What is the truth table for the preceding statement? (4 marks)
Ans:
| X | Y | Z | x ∨ ¬ y ∨ ¬ z |
|---|---|---|---|
| T | T | T | T |
| T | T | F | T |
| T | F | T | T |
| T | F | F | T |
| F | T | T | F |
| F | T | F | T |
| F | F | T | T |
| F | F | F | T |
Ques: Create a truth table for the following statement: (P ∨ Q)∧(~P⇒Q). (4 marks)
Ans:
| P | Q | P ∨ Q | ~P | ~P⇒Q | (P ∨ Q)∧(~P⇒Q) |
|---|---|---|---|---|---|
| T | T | T | F | T | T |
| T | F | T | F | T | T |
| F | T | T | T | T | T |
| F | F | F | T | F | F |
Ques: Write the component statements of the following compound statements and check whether the compound statement is true or false. (i) A line is straight and extends indefinitely in both directions. (ii) 0 is less than every positive integer and every negative integer. (iii) All living things have two legs and two eyes. (3 marks)
Ans: (i) The component statements are p: A line is straight. q: A line extends indefinitely in both directions.
Both these statements are true, therefore, the compound statement is true.
(ii) The component statements are p: 0 is less than every positive integer. q: 0 is less than every negative integer. The second statement is false. Therefore, the compound statement is false.
(iii) The two component statements are p: All living things have two legs. q: All living things have two eyes. Both these statements are false. Therefore, the compound statement is false.
Ques: Determine whether an inclusive "Or" or exclusive "Or" is used in each of the following statements. Give reasons for your response. (4 marks)
I You must have a passport or a voter registration card to enter a country.
(ii) If it is a holiday or a Sunday, the school is closed.
(iii) Two parallel lines or lines that intersect at a point.
(iv) As a third language, students can choose between French and Sanskrit.
Ans: (i)The word "or" is used here because a person can enter a country with both a passport and a voter registration card.
(ii) "Or" is also used here because school is closed on holidays and on Sunday.
(iii) In this case, "Or" is exclusive because two lines cannot intersect and parallel together.
(iv) "Or" is also used here because a student cannot take both French and Sanskrit.
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