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To determine the logical equivalent statement to [tex]\( q \rightarrow p \)[/tex], let's break down what [tex]\( q \rightarrow p \)[/tex] (i.e., "If [tex]\( q \)[/tex], then [tex]\( p \)[/tex]") means in the context of the given propositions:
1. Given propositions:
- [tex]\( p \)[/tex]: Two linear functions have different coefficients of [tex]\( x \)[/tex].
- [tex]\( q \)[/tex]: The graphs of two functions intersect at exactly one point.
2. Restating [tex]\( q \rightarrow p \)[/tex]:
- [tex]\( q \rightarrow p \)[/tex] is read as "If the graphs of two functions intersect at exactly one point, then two linear functions have different coefficients of [tex]\( x \)[/tex]."
To identify a logically equivalent statement, we should understand a few logical concepts:
- Contrapositive: The contrapositive of a statement [tex]\( q \rightarrow p \)[/tex] is [tex]\( \neg p \rightarrow \neg q \)[/tex]. A statement and its contrapositive are logically equivalent.
Let's use this to find a logically equivalent form of [tex]\( q \rightarrow p \)[/tex]:
1. Negation of [tex]\( p \)[/tex]:
- [tex]\( p \)[/tex]: Two linear functions have different coefficients of [tex]\( x \)[/tex].
- [tex]\( \neg p \)[/tex]: Two linear functions have the same coefficients of [tex]\( x \)[/tex].
2. Negation of [tex]\( q \)[/tex]:
- [tex]\( q \)[/tex]: The graphs of two functions intersect at exactly one point.
- [tex]\( \neg q \)[/tex]: The graphs of two functions do not intersect at exactly one point.
3. Forming the contrapositive:
- The contrapositive of [tex]\( q \rightarrow p \)[/tex] is [tex]\( \neg p \rightarrow \neg q \)[/tex].
- So, [tex]\( \neg p \rightarrow \neg q \)[/tex] translates to: "If two linear functions have the same coefficients of [tex]\( x \)[/tex], then the graphs of two functions do not intersect at exactly one point."
Given the options:
- Option 1: If two linear functions have different coefficients of [tex]\( x \)[/tex], then the graphs of the two functions intersect at exactly one point. (This is the original statement [tex]\( p \rightarrow q \)[/tex], not the equivalent of [tex]\( q \rightarrow p \)[/tex]).
- Option 2: If two linear functions have the same coefficients of [tex]\( x \)[/tex], then the graphs of the two linear functions do not intersect at exactly one point. (This is the contrapositive [tex]\( \neg p \rightarrow \neg q \)[/tex], which is logically equivalent to [tex]\( q \rightarrow p \)[/tex]).
- Option 3: If the graphs of two functions do not intersect at exactly one point, then the two linear functions have the same coefficients of [tex]\( x \)[/tex]. (This is [tex]\( \neg q \rightarrow \neg p \)[/tex], which is not the contrapositive of [tex]\( q \rightarrow p \)[/tex]).
- Option 4: If the graphs of two functions intersect at exactly one point, then the two linear functions have the same coefficients of [tex]\( x \)[/tex]. (This is [tex]\( q \rightarrow \neg p \)[/tex], which is not logically equivalent to [tex]\( q \rightarrow p \)[/tex]).
Thus, the correct statement that is logically equivalent to [tex]\( q \rightarrow p \)[/tex] is:
Option 2: If two linear functions have the same coefficients of [tex]\( x \)[/tex], then the graphs of the two linear functions do not intersect at exactly one point.
1. Given propositions:
- [tex]\( p \)[/tex]: Two linear functions have different coefficients of [tex]\( x \)[/tex].
- [tex]\( q \)[/tex]: The graphs of two functions intersect at exactly one point.
2. Restating [tex]\( q \rightarrow p \)[/tex]:
- [tex]\( q \rightarrow p \)[/tex] is read as "If the graphs of two functions intersect at exactly one point, then two linear functions have different coefficients of [tex]\( x \)[/tex]."
To identify a logically equivalent statement, we should understand a few logical concepts:
- Contrapositive: The contrapositive of a statement [tex]\( q \rightarrow p \)[/tex] is [tex]\( \neg p \rightarrow \neg q \)[/tex]. A statement and its contrapositive are logically equivalent.
Let's use this to find a logically equivalent form of [tex]\( q \rightarrow p \)[/tex]:
1. Negation of [tex]\( p \)[/tex]:
- [tex]\( p \)[/tex]: Two linear functions have different coefficients of [tex]\( x \)[/tex].
- [tex]\( \neg p \)[/tex]: Two linear functions have the same coefficients of [tex]\( x \)[/tex].
2. Negation of [tex]\( q \)[/tex]:
- [tex]\( q \)[/tex]: The graphs of two functions intersect at exactly one point.
- [tex]\( \neg q \)[/tex]: The graphs of two functions do not intersect at exactly one point.
3. Forming the contrapositive:
- The contrapositive of [tex]\( q \rightarrow p \)[/tex] is [tex]\( \neg p \rightarrow \neg q \)[/tex].
- So, [tex]\( \neg p \rightarrow \neg q \)[/tex] translates to: "If two linear functions have the same coefficients of [tex]\( x \)[/tex], then the graphs of two functions do not intersect at exactly one point."
Given the options:
- Option 1: If two linear functions have different coefficients of [tex]\( x \)[/tex], then the graphs of the two functions intersect at exactly one point. (This is the original statement [tex]\( p \rightarrow q \)[/tex], not the equivalent of [tex]\( q \rightarrow p \)[/tex]).
- Option 2: If two linear functions have the same coefficients of [tex]\( x \)[/tex], then the graphs of the two linear functions do not intersect at exactly one point. (This is the contrapositive [tex]\( \neg p \rightarrow \neg q \)[/tex], which is logically equivalent to [tex]\( q \rightarrow p \)[/tex]).
- Option 3: If the graphs of two functions do not intersect at exactly one point, then the two linear functions have the same coefficients of [tex]\( x \)[/tex]. (This is [tex]\( \neg q \rightarrow \neg p \)[/tex], which is not the contrapositive of [tex]\( q \rightarrow p \)[/tex]).
- Option 4: If the graphs of two functions intersect at exactly one point, then the two linear functions have the same coefficients of [tex]\( x \)[/tex]. (This is [tex]\( q \rightarrow \neg p \)[/tex], which is not logically equivalent to [tex]\( q \rightarrow p \)[/tex]).
Thus, the correct statement that is logically equivalent to [tex]\( q \rightarrow p \)[/tex] is:
Option 2: If two linear functions have the same coefficients of [tex]\( x \)[/tex], then the graphs of the two linear functions do not intersect at exactly one point.
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