Question

“If natural numbers are whole numbers, then rational numbers are integers” or “If rational numbers are integers, then natural numbers are whole numbers” is
A. A tautology
B. a contradiction
C. a contingency
D. not a statement

Answer 1

Hint:
A tautology, or tautologous proposition, has a logical form that cannot possibly be false (no matter what truth values are assigned to the sentence letters). So, this means that the final logic for the statement should be true to become a tautology. In the above statements, both the statements are false and hence ultimately they logically represent true. Hence, we will get the answer.

Complete step by step solution:
Introduction to propositional logic
A proposition is a statement that can either be true or false. Whenever we consider two different propositions together, can we say that the overall compound proposition is true or false? To answer this propositional logic is formed. It consists of these important terms:

Tautology
A tautology is said when a compound proposition is always TRUE. i.e The overall logic of the statement is True. We can say that if we make a truth table for the propositional composition, we will always get ‘true’ as the output no matter what the input be.

Contradiction
A Contradiction is said when a compound proposition is always FALSE. i.e The overall logic of the statement is false. We can say that if we make a truth table for the propositional composition, we will always get ‘false’ as the output no matter what the input be.

Contingency
A Contingency is said when the compound proposition may be TRUE or FALSE. The overall logic of the statement here may be true or false. We can say that if we make a truth table for the propositional composition, we may get true or false as the output no matter what the input be.

Now, look again at the question,
Let,
P = “If natural numbers are whole numbers, then rational numbers are integers”
Q = “If rational numbers are integers, then natural numbers are whole numbers”

In the given proposition we can easily see that P and Q both are false, hence the compound preposition would be TRUE.

Hence, the final answer is A.

Note:
There are two more important terms which fall under propositional logic
Satisfiability
Those propositional logic whose truth table has at least one output as TRUE is considered to be Satisfiable. So, a tautology and a contingency are satisfiable.
Validity
Those propositional logic whose truth table has not even one FALSE as output is considered to be Valid. Only, a tautology is valid according to this. Contradiction and Contingency are invalid.