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A number is a rational number if and only if it can be represented as a terminating decimal.

Let:
p: A number is a rational number.
q: A number can be represented as a terminating decimal.

Which represents the converse of this statement? Is the converse true?

A. [tex]$\sim q \leftrightarrow \sim p$[/tex]
B. The converse of the statement is false.
C. [tex]$\sim p \leftrightarrow \sim q$[/tex]
D. The converse of the statement is sometimes true and sometimes false.
E. The converse of the statement is true.
F. [tex]$q \leftrightarrow p$[/tex]
G. [tex]$p \leftrightarrow q$[/tex]

Sagot :

GinnyAnswer
To tackle this problem, we'll analyze the given logical statements and their converses.

### Original Statement:
The original statement provided is:
"A number is a rational number if and only if it can be represented as a terminating decimal."

Using the given notation:
- [tex]\( p \)[/tex]: A number is a rational number.
- [tex]\( q \)[/tex]: A number can be represented as a terminating decimal.

This original statement can be written as:
[tex]\[ p \leftrightarrow q \][/tex]
which means: "A number is a rational number if and only if it can be represented as a terminating decimal."

### Converse of the Statement:
The converse of the statement is obtained by swapping [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
[tex]\[ q \leftrightarrow p \][/tex]

In plain language, the converse statement is:
"A number can be represented as a terminating decimal if and only if it is a rational number."

Now, let's examine the truth of the converse:
- The original statement itself is somewhat misleading because not all rational numbers can be represented as terminating decimals. Rational numbers can also be represented as repeating (non-terminating) decimals (for example, [tex]\( \frac{1}{3} = 0.333... \)[/tex]).

However, every terminating decimal can be expressed as a rational number because it can be written as a fraction of two integers.

This means:
- The converse statement "[tex]$q \leftrightarrow p$[/tex]" is true interpretatively because every terminating decimal is, in fact, a rational number.

### Analysis of Given Options:
- [tex]\(\sim q \leftrightarrow \sim p\)[/tex]: This represents the contrapositive form, which is logically equivalent to the original statement but not the converse.
- The converse of the statement is false: This is incorrect because the converse can be true as discussed.
- [tex]\(\sim p \leftrightarrow \sim q\)[/tex]: This is also the contrapositive and not the converse.
- The converse of the statement is sometimes true and sometimes false: Misleading, because as interpreted correctly, the converse is true.
- The converse of the statement is true: This is contextually accurate.
- [tex]\(q \leftrightarrow p\)[/tex]: This correctly represents the converse statement.
- [tex]\(p \leftrightarrow q\)[/tex]: This is the original statement, not the converse.

### Conclusion:
From the options, the correct answers are:
- The converse of the statement is true.
- [tex]\(q \leftrightarrow p\)[/tex]

Thus, the correct selections are:
1. The converse of the statement is true.
2. [tex]\(q \leftrightarrow p\)[/tex]
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