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Truth Tables

Truth table is a table showing the truth value of a statement formula for each possible combinations of truth values of component statements. A statement formula having n distinct components will have 2$^{2}$ rows in its truth table. A statement is a declarative sentence which has one and only one of the two possible values, called truth values. These two truth values are true and false denoted by the symbols T and F respectively, sometimes also denoted by symbols 1 and 0. Since we allow only two possible truth values, this logic is called two-valued logic.

Truth table
is a powerful concepts that constructs truth tables for its component statements. It is the most preferred tool in Boolean algebra. Truth table is a mathematical table specifically in connection with Boolean algebra, Boolean functions and propositional calculus. A mathematical table that displays all the possible truth values of a logical operation, is known as a truth table. It computes the functional values of logical expressions on each of their functional arguments.

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Truth Tables For Unary Operations

The unary logical operations are those operations which contains only one logical variable.

Truth Table for Logical True

Logical true returns a true value for whatever every input. Its truth table is :

 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 F(P) T F F F

Truth table for negation

Logical negation is a unary operation which typically returns opposite value of a proposition. If input is true, then output is false and vice versa. It is represented by NOT, or  ¬p or Np or ~p. The truth table for NOT is given below.

Truth Tables For Binary Operations

In logical mathematics, the binary operations are the logical operations that have two logical input variables. The truth tables of most important binary operations are given below.

Truth Table For Conjunction

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 $\wedge$ Q or P . Q or P & Q, where P and Q are input variables. Its truth table is given below :

 P Q (P$\wedge$Q) And 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 is true, i.e. either one of them or both are true.
It is denoted by the symbols P OR Q, P $\vee$ Q or P + Q. Its truth table is shown below.

 P Q (P$\vee$Q) or 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 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 and is denoted by P $\rightarrow$ Q or P $\Rightarrow$ Q. The truth table for implication is as follows:

 P Q (P $\rightarrow$ Q) T T T T F F F T T F F T

We can learn another logic operations with truth tables.The propositional logic truth tables are standard one.So we can't change the propositional value.

Logical NAND

The NAND is a binary logical operation which similar to applying NOT on AND operation. In other words, NAND produces a true value if at least one of 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$\wedge$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$\vee$Q) or T T F T F F F T F F F T

Examples

The students can understand the propositional logic truth tables by using given example problems and they should practice the exercise problems.

Example 1: Find out the logical truth tables for given value using implication.

Input A: T T T F F T

Input B: T F F T T T

Solution:

 A B Propositional value(P$\rightarrow$Q) T T T T F F T F F F T T F T T T T T

The propositional A $\rightarrow$ B is T F F T T T.

Example 2: Find out the logical truth table for given values using conjunction.

Input A: F F T F T

Input B: F T T T F

Solution:

 A B Propositional value(P$\wedge$Q) F F F F T F T T T F T F T F F

The propositional A $\wedge$ B is F F T F F.

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