Answer :
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]
### 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]
