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Sagot :
To determine which statement is logically equivalent to the conditional statement [tex]\( p \rightarrow q \)[/tex], let's analyze the given options step-by-step and recall some fundamental concepts of logic.
1. Original Statement: [tex]\( p \rightarrow q \)[/tex]
- This reads as "if [tex]\( p \)[/tex] then [tex]\( q \)[/tex]."
2. Negation ([tex]\(\sim\)[/tex]):
- The negation of a statement [tex]\( p \)[/tex] is [tex]\(\sim p\)[/tex], meaning "not [tex]\( p \)[/tex]."
3. Contrapositive:
- The contrapositive of the statement [tex]\( p \rightarrow q \)[/tex] is [tex]\(\sim q \rightarrow \sim p\)[/tex]. This reads as "if not [tex]\( q \)[/tex] then not [tex]\( p \)[/tex]."
- It is well-known that a conditional statement is logically equivalent to its contrapositive. Thus, [tex]\( p \rightarrow q \)[/tex] is logically equivalent to [tex]\(\sim q \rightarrow \sim p\)[/tex].
Now, let's examine all the given options in relation to [tex]\( p \rightarrow q \)[/tex]:
1. Option 1: [tex]\(\sim p \rightarrow \sim q\)[/tex]
- This reads as "if not [tex]\( p \)[/tex] then not [tex]\( q \)[/tex]."
- This is NOT logically equivalent to the original statement [tex]\( p \rightarrow q \)[/tex].
2. Option 2: [tex]\(\sim q \rightarrow \sim p\)[/tex]
- This reads as "if not [tex]\( q \)[/tex] then not [tex]\( p \)[/tex]."
- As established earlier, this is the contrapositive of the original statement [tex]\( p \rightarrow q \)[/tex] and is logically equivalent to it.
3. Option 3: [tex]\( q \rightarrow p \)[/tex]
- This reads as "if [tex]\( q \)[/tex] then [tex]\( p \)[/tex]."
- This is the converse of the original statement [tex]\( p \rightarrow q \)[/tex] and is NOT logically equivalent to it.
4. Option 4: [tex]\( p \rightarrow \sim q \)[/tex]
- This reads as "if [tex]\( p \)[/tex] then not [tex]\( q \)[/tex]."
- This is simply another conditional statement and is NOT logically equivalent to [tex]\( p \rightarrow q \)[/tex].
Therefore, based on the logical analysis above, the statement [tex]\( \sim q \rightarrow \sim p \)[/tex] is the one that is logically equivalent to [tex]\( p \rightarrow q \)[/tex]. Hence, the correct choice is:
[tex]\[ \boxed{\sim q \rightarrow \sim p} \][/tex]
This corresponds to option 2.
1. Original Statement: [tex]\( p \rightarrow q \)[/tex]
- This reads as "if [tex]\( p \)[/tex] then [tex]\( q \)[/tex]."
2. Negation ([tex]\(\sim\)[/tex]):
- The negation of a statement [tex]\( p \)[/tex] is [tex]\(\sim p\)[/tex], meaning "not [tex]\( p \)[/tex]."
3. Contrapositive:
- The contrapositive of the statement [tex]\( p \rightarrow q \)[/tex] is [tex]\(\sim q \rightarrow \sim p\)[/tex]. This reads as "if not [tex]\( q \)[/tex] then not [tex]\( p \)[/tex]."
- It is well-known that a conditional statement is logically equivalent to its contrapositive. Thus, [tex]\( p \rightarrow q \)[/tex] is logically equivalent to [tex]\(\sim q \rightarrow \sim p\)[/tex].
Now, let's examine all the given options in relation to [tex]\( p \rightarrow q \)[/tex]:
1. Option 1: [tex]\(\sim p \rightarrow \sim q\)[/tex]
- This reads as "if not [tex]\( p \)[/tex] then not [tex]\( q \)[/tex]."
- This is NOT logically equivalent to the original statement [tex]\( p \rightarrow q \)[/tex].
2. Option 2: [tex]\(\sim q \rightarrow \sim p\)[/tex]
- This reads as "if not [tex]\( q \)[/tex] then not [tex]\( p \)[/tex]."
- As established earlier, this is the contrapositive of the original statement [tex]\( p \rightarrow q \)[/tex] and is logically equivalent to it.
3. Option 3: [tex]\( q \rightarrow p \)[/tex]
- This reads as "if [tex]\( q \)[/tex] then [tex]\( p \)[/tex]."
- This is the converse of the original statement [tex]\( p \rightarrow q \)[/tex] and is NOT logically equivalent to it.
4. Option 4: [tex]\( p \rightarrow \sim q \)[/tex]
- This reads as "if [tex]\( p \)[/tex] then not [tex]\( q \)[/tex]."
- This is simply another conditional statement and is NOT logically equivalent to [tex]\( p \rightarrow q \)[/tex].
Therefore, based on the logical analysis above, the statement [tex]\( \sim q \rightarrow \sim p \)[/tex] is the one that is logically equivalent to [tex]\( p \rightarrow q \)[/tex]. Hence, the correct choice is:
[tex]\[ \boxed{\sim q \rightarrow \sim p} \][/tex]
This corresponds to option 2.
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