Answer :
To determine which statement is logically equivalent to [tex]\(q \rightarrow p\)[/tex], we first need to clearly understand the logical implications of the given statements.
First, let's break down [tex]\(q\)[/tex] and [tex]\(p\)[/tex]:
- [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.
The implication [tex]\(q \rightarrow p\)[/tex] translates to: "If the graphs of two functions intersect at exactly one point, then the two linear functions have different coefficients of [tex]\(x\)[/tex]."
To find the logically equivalent statement, we should look for a statement that conveys the same logical relationship. One approach is to identify the contrapositive of [tex]\(q \rightarrow p\)[/tex]. The contrapositive of a statement [tex]\(A \rightarrow B\)[/tex] is [tex]\(\neg B \rightarrow \neg A\)[/tex], and it is logically equivalent to the original statement.
So, the contrapositive of [tex]\(q \rightarrow p\)[/tex] is:
1. [tex]\(\neg p\)[/tex]: The two linear functions do not have different coefficients of [tex]\(x\)[/tex] (i.e., the two linear functions have the same coefficients of [tex]\(x\)[/tex]).
2. [tex]\(\neg q\)[/tex]: The graphs of the two functions do not intersect at exactly one point.
Putting it together, the contrapositive of [tex]\(q \rightarrow p\)[/tex] is: "If the two linear functions have the same coefficients of [tex]\(x\)[/tex], then the graphs of the two functions do not intersect at exactly one point."
Now let's compare this statement with the given options:
1. If two linear functions have different coefficients of [tex]\(x\)[/tex], then the graphs of the two functions intersect at exactly one point.
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.
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].
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].
The correct answer 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."
This statement is the contrapositive of [tex]\(q \rightarrow p\)[/tex] and is therefore logically equivalent to the original statement.
First, let's break down [tex]\(q\)[/tex] and [tex]\(p\)[/tex]:
- [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.
The implication [tex]\(q \rightarrow p\)[/tex] translates to: "If the graphs of two functions intersect at exactly one point, then the two linear functions have different coefficients of [tex]\(x\)[/tex]."
To find the logically equivalent statement, we should look for a statement that conveys the same logical relationship. One approach is to identify the contrapositive of [tex]\(q \rightarrow p\)[/tex]. The contrapositive of a statement [tex]\(A \rightarrow B\)[/tex] is [tex]\(\neg B \rightarrow \neg A\)[/tex], and it is logically equivalent to the original statement.
So, the contrapositive of [tex]\(q \rightarrow p\)[/tex] is:
1. [tex]\(\neg p\)[/tex]: The two linear functions do not have different coefficients of [tex]\(x\)[/tex] (i.e., the two linear functions have the same coefficients of [tex]\(x\)[/tex]).
2. [tex]\(\neg q\)[/tex]: The graphs of the two functions do not intersect at exactly one point.
Putting it together, the contrapositive of [tex]\(q \rightarrow p\)[/tex] is: "If the two linear functions have the same coefficients of [tex]\(x\)[/tex], then the graphs of the two functions do not intersect at exactly one point."
Now let's compare this statement with the given options:
1. If two linear functions have different coefficients of [tex]\(x\)[/tex], then the graphs of the two functions intersect at exactly one point.
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.
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].
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].
The correct answer 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."
This statement is the contrapositive of [tex]\(q \rightarrow p\)[/tex] and is therefore logically equivalent to the original statement.
