Answer:
The answer is: the satement is FALSE.
Step-by-step explanation:
The first step is to transform the propositons in the statement into symbols:
A: m is any positive integer
B: n is any positive integer
C: mn is a a perfect square
D: m is a perfect square
E: n is a perfect square
After that we replace the symbols into the statement:
If A and B and C, then D and E
Logical connectives on the satement are:
- p⇒q conditional (if ...,then...). For this connective the only case when is FALSE is when the antecedent is true and the consequent is false.
- p ∧ q conjunction (p and q). For this connective the only case when is TRUE is when the two propositions are true at the same time.
So now, we are going to replace the logical connectives on the statement:
((A ∧ B) ∧ C) ⇒ (D ∧ E)
There is a hierarchy in logical connectives, first we evaluate those inside brackets. The principal connective is evaluated at the end. In this particular case, the principal connective is the conditional.
As we have five different propositions, the combinations are given by the following rule:
2^n= 2^5=32
32 different combinations
In figure 1, added bellow, it shows the construction of truth table.
In figure 2, added bellow, it shows the development of the truth table. In the development, we don't have as a result a tautology, therefore the statement is FALSE.
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