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A truth table is a mathematical table used in logic to list the functional values of logical expressions on each of their functional arguments, or each set of values taken by their logical variables. Truth tables can be used to show if a prepositional phrase is true for all logically valid input values.
Definition of a Truth Table
A truth table is a mathematical table that determines if a compound assertion is true or not. Each statement in a truth table is commonly denoted by a letter or variable, such as p, q, or r, and each statement has its own column in the truth table containing all of the possible truth values.
Unary Operations Truth Tables
The notion of a truth table is a powerful one that creates truth tables for its component claims. The unary logical operations, on the other hand, are those that have only one logical variable. Let’s take a closer look.
Logical True Truth Table
For every input, logical true returns a true value. Its truth table is as follows:
P | T(P) | P |
T | T | T |
F | T | F |
Logical False Truth Table
Whatever the input is, logical false returns a false value. The following is its truth table.
P | T(P) | P |
T | F | T |
F | F | F |
Negation’s Truth Table
A negation’s probable truth values are the inverse of the truth values of the assertion it denies. If p is true,∼p must be false. When p is false, ∼p is true.
P | ∼P | P |
T | F | T |
F | T | F |
Below are the Truth Tables for the Most Important Binary Operations are Given Below
Conjunction Truth Table
A conjunction p ∧ q is true only when both of its conjuncts are true. In the other three circumstances, it is incorrect.
P | Q | P ∧ Q |
T | T | T |
T | F | F |
F | T | F |
F | F | F |
Disjunction Truth Table
Only when both of its disjuncts are untrue is a disjunction p ∨ q false. The disjunction is true in the other three circumstances:
P | Q | P ∨ Q |
T | T | T |
T | F | T |
F | T | T |
F | F | F |
Implications Truth Table
When the first input is true and the second is either false or true, logical implication usually returns a value of false. It is associated with the condition, “if P then Q” and is denoted by P → Q or P ⇒ Q. The truth table for inference is as follows:
P | Q | P → Q |
T | T | T |
T | F | F |
F | T | T |
F | F | T |
Tables of Logic and Truth
The logic tables operations using truth tables will be discussed here. The truth tables for propositional logic are the most common. As a result, we can’t alter the propositional value.
Logical NAND
NAND is a binary logical operation that works in the same way as NOT on AND. In other words, if at least one of the input variables is false, NAND returns a true value. t is denoted by P NAND Q or P | Q or P ↑ Q. Take a look at the truth table it contains.
P | Q | (P ∧ Q) And |
T | T | F |
T | F | T |
F | T | T |
F | F | T |
Logical NOR
A logical NOR is a logical operation which is obtained by applying a NOT operation to an OR operation. If both input variables are false, we can argue that NOR produces a true value. P NOR Q or P ↓Q are the symbols for it. Take a look at the truth table it provides.
P | Q | (P ∨ Q) OR |
T | T | F |
T | F | F |
F | T | F |
F | F | T |
Steps to Construct a Truth table
Step 1: Recognize truth tables.
Step 2: Symbol recognition
Step 4: Assign true and false to each other.
Step 5: Negation
Step 6: Variable “q”
Step 7: Fake in the last column solution
Step 8: Look in the last column for the True.
Step 9: Complete the Table
Step 10: Completed
Examples of Truth Tables
Example 1: Using conjunction, find the logical truth table for given values.
If P is F T F T and Q is F T T F, then
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 |
Conclusion
The truth table is a mathematical table that breaks down a logical function by stating all of the values that the function can achieve. The truth table of logic gates contains all of the information regarding the combination of inputs and outputs for a logic operation.
