Answer
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Hint: We know that a tautology is a formula which is "always true" that is, it is true for every assignment of truth values to its simple components. You can think of a tautology as a rule of logic. The opposite of a tautology is a contradiction, a formula which is "always false". A proposition which is neither a tautology nor a contradiction is called contingency. Write a truth table for the given proposition and see whether all are true or all are false or neither of those.
Complete step by step answer:
By first writing the truth table of $(p \Rightarrow p)$, we get it as
Now by writing the truth table of $(p \Rightarrow p) \wedge (p \Rightarrow p)$, we get it as
Truth table for a tautology has T in its every row.
Truth table for a contradiction has F in its every row.
A proposition which is neither a tautology nor a contradiction is called contingency.
Contingency has both T and F in its truth table.
In this resulted table we can see that all the outputs are T i.e. True.
Therefore, it is clearly tautology because all the outputs are T irrespective of the value of p.
So, the correct answer is option A.
Note:
Read the definitions of tautology , contradiction and contingency. Practice more problems to get a hold of these types of problems. Remember that if a composite proposition is contingent then it cannot be tautology and it also cannot be a contradiction.
Complete step by step answer:
By first writing the truth table of $(p \Rightarrow p)$, we get it as
p | p | $(p \Rightarrow p)$ |
T | T | T |
F | F | T |
Now by writing the truth table of $(p \Rightarrow p) \wedge (p \Rightarrow p)$, we get it as
p | p | $(p \Rightarrow p)$ | $(p \Rightarrow p) \wedge (p \Rightarrow p)$ |
T | T | T | T |
F | F | T | T |
Truth table for a tautology has T in its every row.
Truth table for a contradiction has F in its every row.
A proposition which is neither a tautology nor a contradiction is called contingency.
Contingency has both T and F in its truth table.
In this resulted table we can see that all the outputs are T i.e. True.
Therefore, it is clearly tautology because all the outputs are T irrespective of the value of p.
So, the correct answer is option A.
Note:
Read the definitions of tautology , contradiction and contingency. Practice more problems to get a hold of these types of problems. Remember that if a composite proposition is contingent then it cannot be tautology and it also cannot be a contradiction.
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