(a.)
To Show: The given expression is a tautology, using truth table.
It has been determined that the given expression is a tautology.
Given:
Concept used:
If the truth value of an expression in its truth table is always true, then it is a tautology.
Calculation:
The given expression is
The truth table for the given expression is as follows:
| F | F | F | T | F | T | T |
| F | F | T | T | F | T | T |
| F | T | F | T | F | T | T |
| F | T | T | T | F | T | T |
| T | F | F | F | F | T | T |
| T | F | T | F | T | F | T |
| T | T | F | T | F | T | T |
| T | T | T | T | T | T | T |
It can be seen that the truth value for the given expression is always true.
Hence, the given expression is a tautology.
Conclusion:
It has been determined that the given expression is a tautology.
(b.)
To Show: The given expression is a tautology, using truth table.
It has been determined that the given expression is a tautology.
Given:
Concept used:
If the truth value of an expression in its truth table is always true, then it is a tautology.
Calculation:
The given expression is
The truth table for the given expression is as follows:
| F | F | T | F | T |
| F | T | T | F | T |
| T | F | F | F | T |
| T | T | T | T | T |
It can be seen that the truth value for the given expression is always true.
Hence, the given expression is a tautology.
Conclusion:
It has been determined that the given expression is a tautology.
(c.)
To Show: The given expression is a tautology, using truth table.
It has been determined that the given expression is a tautology.
Given:
Concept used:
If the truth value of an expression in its truth table is always true, then it is a tautology.
Calculation:
The given expression is
The truth table for the given expression is as follows:
| F | F | T | T | T | T |
| F | T | F | T | F | T |
| T | F | T | F | F | T |
| T | T | F | T | F | T |
It can be seen that the truth value for the given expression is always true.
Hence, the given expression is a tautology.
Conclusion:
It has been determined that the given expression is a tautology.
(d.)
To Show: The given expression is a tautology, using truth table.
It has been determined that the given expression is a tautology.
Given:
Concept used:
If the truth value of an expression in its truth table is always true, then it is a tautology.
Calculation:
The given expression is
The truth table for the given expression is as follows:
| F | F | F | T | T | T | T | T |
| F | F | T | T | T | T | T | T |
| F | T | F | T | F | F | T | T |
| F | T | T | T | T | T | T | T |
| T | F | F | F | T | F | F | T |
| T | F | T | F | T | F | T | T |
| T | T | F | T | F | F | F | T |
| T | T | T | T | T | T | T | T |
It can be seen that the truth value for the given expression is always true.
Hence, the given expression is a tautology.
Conclusion:
It has been determined that the given expression is a tautology.
Chapter B.2 Solutions
PRECALCULUS:GRAPHICAL,...-NASTA ED.
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