Truth Tables
In Boolean algebra, truth table is a table showing the truth value of a statement formula for each possible combination of truth values of component statements. A statement is a declarative sentence which has one and only one of the two possible values called truth values. Truth values are true and false, denoted by the symbols T and F, respectively, and they are also denoted by symbols 1 and 0. Since we allow only two possible truth values, this logic is called two-valued logic.
Truth Tables for Unary Operations
The truth table is a powerful concept that constructs truth tables for its component statements. On the other hand, unary logical operations are those operations which contain only one logical variable. Let’s study this in detail.
Truth Table for Logical True
Logical true returns a true value for every input. Its truth table is as follows:
| P | T(P) |
| T | T |
| F | T |
Truth Table for Logical False
Logical false gives a false value for whatever the input is. Its truth table is as follows:
| P | T(P) |
| T | F |
| F | F |
Truth Table for Negation
Logical negation is a unary operation which typically returns the opposite value of a proposition. If the input is false, then the output is true and vice versa. It is represented by NOT or ~p. The truth table for NOT is given below.
| P | ∼P |
| T | F |
| F | T |
Truth Tables for Binary Operations
In logical mathematics, binary operations are logical operations that have two logical input variables. The truth tables of the most important binary operations are given below.
Truth Table for Conjunction
A conjunction is a binary logical operation which results in a true value if both the input variables are true. This operator is represented by P AND Q or P ∧ Q or P . Q or P & Q, where P and Q are input variables. Its truth table is given below:
| P | Q | P ∧ Q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Truth Table for Disjunction
Logical disjunction returns a true when at least input operands are true, i.e., either one of them or both are true.
It is denoted by the symbols P OR Q, P ∨ Q or P + Q. Its truth table is shown below:
| P | Q | P ∨ Q |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Truth Table for Implication
Logical implication typically produces a value of false in the singular case that the first input is true and the second is either false or true. It is associated with the condition, “if P then Q” [Conditional Statement] and is denoted by P → Q or P ⇒ Q. The truth table for implication is as follows:
| P | Q | P → Q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Biconditional Truth Table
It is associated with the condition, “P if and only if Q” [Biconditional Statement] and is denoted by:
| P | Q | \(\begin{array}{l}P\leftrightarrow Q\end{array} \) |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Logic and Truth Tables
Here, we will discuss the logic tables operations with truth tables. The propositional logic truth tables are the standard ones. So, we can’t change the propositional value.
Logical NAND
The NAND is a binary logical operation which is similar to applying NOT on AND operation. In other words, NAND produces a true value if at least one of the input variables is false. It is denoted by P NAND Q or P | Q or P ↑ Q. Have a look at its truth table.
| P | Q | (P ∧ Q) And |
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | T |
Logical NOR
The logical NOR is a logical operation which is obtained by applying a NOT operation to an OR operation. We can say that NOR results in a true value if both the input variables are false. It is represented by P NOR Q or P ↓ Q. Take a look at its truth table.
| P | Q | (P ∨ Q) OR |
| T | T | F |
| T | F | F |
| F | T | F |
| F | F | T |
Steps to Solve Logical Expressions
To solve logical expressions, we proceed as follows:
1. First, solve the parentheses, if any.
2. Next, solve the NOT operator.
3. Then, the AND operator.
4. And lastly, the OR operator.
Logical Thinking and Statements
Logical thinking is the ability to understand the facts and concepts of the quantity, ideas, number system and other relationships between the ideas and the concepts. It is an ability to view a concept in a different method. It is a good skill in maths problem-solving. It deals with the ability to solve mathematical problems, planning, predicting outcomes, voluntary risk etc.
The standards of logical-mathematical intelligence concepts
The ability for Problem-solving
Skills of Communication
Ability of Reasoning
Viewing Geometry and Spatial Sense
Methods of Measurement
Statistics and Probability
Patterns and Relationships, and many more
Important Laws Followed by Statements
If A, B and C are statements, then
| Commutative Laws | A ∨ B = B ∨ A and A ∧ B = A ∧ B |
| Associative Laws | A ∨ (B ∨ C) = (A ∨ B) ∨ C and
A ∧ (B ∧ C) = (A ∧ B) ∧ C |
| Distributive Laws | A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C) and
A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C) |
| Idempotent Laws | A ∨ B = A and
A ∧ B = A |
| Identity Laws | A ∨ t = t,
A ∧ t = A, A ∨ m = A and A ∧ m = m |
| Complement Laws | A ∨ (~A) = t,
A ∧ (~A) = m, ~t = m and ~m = t |
| De-Morgan’s Law | ~(A ∨ B) = (~A) ∧ (~B) and
~(A ∧ B) = (~A) ∨ (~B) |
Truth Tables Examples
Example 1: Find the logical truth table for given values using conjunction.
If P: F F T F T and Q: F T T T F
Solution:
| P | Q | P ∧ Q |
| F | F | F |
| F | T | F |
| T | T | T |
| F | T | F |
| T | F | F |
Example 2: Construct the truth table for ~P∨∼Q and ∼(P∧Q).
Solution:
| P | Q | ~P | ~Q | ~P∨∼Q | (P∧Q) | ~(P∧Q) |
| T | T | F | F | F | T | F |
| T | F | F | T | T | F | T |
| F | T | T | F | T | F | T |
| F | F | T | T | T | F | T |
Example 3: Which of the following is the contrapositive of ‘if two triangles are identical, then these are similar’?
A) If two triangles are not similar, then they are not identical.
B) If two triangles are not identical, then these are not similar.
C) If two triangles are not identical, then these are similar.
D) If two triangles are not similar, then these are identical.
Solution:
Consider the following statements
p: Two triangles are identical.
q: Two triangles are similar.
Clearly, the given statement in symbolic form is p ⇒ q.
Therefore, its contrapositive is given by ∼q ⇒ ∼p
Now,
Example 4: The statement ~(p↔ ~q) is
A) equivalent to p↔q
B) equivalent to ~p↔q
C) a tautology
D) a fallacy
Solution:
| p | q | ∼q | p↔∼q | ∼(p↔∼q) | p↔q |
| T | T | F | F | T | T |
| T | F | T | T | F | F |
| F | T | F | T | F | F |
| F | F | T | F | T | T |
Example 5: Let S be a non-empty subset of R. Consider the following statement: P: There is a rational number x ∈ S such that x>0. Which of the following statements is the negation of the statement p?
A) x ∈ S and x ≤ 0 ⇒ x is not rational.
B) There is a rational number x ∈ S such that x ≤ 0.
C) There is no rational number x ∈ S such that x ≤ 0.
D) Every rational number x ∈ S satisfies x ≤ 0.
Solution:
P: There is a rational number x ∈ S such that x > 0.
~ P: Every rational number x ∈ S satisfies x ≤ 0.
Also, Read:
Video on Conditional Statements
Frequently Asked Questions
What is a statement in truth table?
A statement is a sentence or mathematical expression which is either definitely true or definitely false but not both.
Define a truth table in Mathematics.
A truth table is a chart of rows and columns showing the truth value (either “T” for True or “F” for False) of every possible combination of the given statements as operated by logical connectives.
What is meant by the negation of a statement?
The negation of a statement is the opposite of the given mathematical statement. It is denoted by the ~ symbol. If “P” is a statement, then the negation of statement P is ~P.
Let P: 5 is greater than 3.
Then ~P: 5 is not greater than 3.
What is meant by conjunction in Mathematics?
A conjunction is a statement formed by adding two statements with the connector AND. The symbol for conjunction is ‘∧’, and it is read as ‘and’.
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