Get detailed and accurate responses to your questions with IDNLearn.com. Ask anything and receive immediate, well-informed answers from our dedicated community of experts.
Sagot :
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.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Find the answers you need at IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.
