Answer :
To determine the correct inverse of the proposition [tex]\( p \rightarrow q \)[/tex], we need to check various logical scenarios involving [tex]\( p \)[/tex] and [tex]\( q \)[/tex].
Let's begin by defining our propositions:
- [tex]\( p \)[/tex]: [tex]\( x - 5 = 10 \)[/tex]
- [tex]\( q \)[/tex]: [tex]\( 4x + 1 = 61 \)[/tex]
First, let's find the values of [tex]\( x \)[/tex] that satisfy [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
1. Solving for [tex]\( p \)[/tex]:
[tex]\[ x - 5 = 10 \][/tex]
[tex]\[ x = 15 \][/tex]
2. Solving for [tex]\( q \)[/tex]:
[tex]\[ 4x + 1 = 61 \][/tex]
[tex]\[ 4x = 60 \][/tex]
[tex]\[ x = 15 \][/tex]
Both propositions are true if [tex]\( x = 15 \)[/tex].
Next, consider the statement [tex]\( p \rightarrow q \)[/tex]. This implication is logically equivalent to [tex]\( \neg p \lor q \)[/tex] (not [tex]\( p \)[/tex] or [tex]\( q \)[/tex]).
Now, let's determine the truth values for different conditions:
1. If [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is true:
From the above solutions, we know that when [tex]\( x = 15 \)[/tex], both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are true:
[tex]\[ p: 15 - 5 = 10 \quad \text{(true)} \][/tex]
[tex]\[ q: 4 \cdot 15 + 1 = 61 \quad \text{(true)} \][/tex]
2. If [tex]\( x - 5 \neq 10 \)[/tex], then [tex]\( 4x + 1 \neq 61 \)[/tex]:
This statement would require that when [tex]\( p \)[/tex] is false ([tex]\( x \neq 15 \)[/tex]), then [tex]\( q \)[/tex] would also be false. However:
- Suppose [tex]\( x = 16 \)[/tex], then [tex]\( x - 5 = 11 \)[/tex] (false for [tex]\( p \)[/tex]), but [tex]\( 4 \cdot 16 + 1 = 65 \)[/tex] (false for [tex]\( q \)[/tex]).
- Suppose [tex]\( x = 14 \)[/tex], then [tex]\( x - 5 = 9 \)[/tex] (false for [tex]\( p \)[/tex]), but [tex]\( 4 \cdot 14 + 1 = 57 \)[/tex] (false for [tex]\( q \)[/tex]).
3. If [tex]\( 4x + 1 \neq 61 \)[/tex], then [tex]\( x - 5 \neq 10 \)[/tex]:
This statement requires that when [tex]\( q \)[/tex] is false, [tex]\( p \)[/tex] must also be false. Again, examining:
- If [tex]\( x = 16 \)[/tex], [tex]\( q \)[/tex] is false ([tex]\( 4 \cdot 16 + 1 = 65 \)[/tex]), and [tex]\( p \)[/tex] is false ([tex]\( x - 5 = 11 \)[/tex]).
- If [tex]\( x = 14 \)[/tex], [tex]\( q \)[/tex] is false ([tex]\( 4 \cdot 14 + 1 = 57 \)[/tex]), and [tex]\( p \)[/tex] is false ([tex]\( x - 5 = 9 \)[/tex]).
4. If [tex]\( x - 5 = 10 \)[/tex], then [tex]\( 4x + 1 = 61 \)[/tex]:
From the values of [tex]\( x = 15 \)[/tex], if [tex]\( p \)[/tex] is true, we see:
[tex]\[ p: 15 - 5 = 10 \quad \text{(true)} \][/tex]
[tex]\[ q: 4 \cdot 15 + 1 = 61 \quad \text{(true)} \][/tex]
5. If [tex]\( 4x + 1 = 61 \)[/tex], then [tex]\( x - 5 = 10 \)[/tex]:
Similarly, if [tex]\( q \)[/tex] is true (with [tex]\( x = 15 \)[/tex]), then:
[tex]\[ q: 4 \cdot 15 + 1 = 61 \quad \text{(true)} \][/tex]
[tex]\[ p: 15 - 5 = 10 \quad \text{(true)} \][/tex]
Given all of these conditions, the inverses of [tex]\( p \rightarrow q \)[/tex] are checked. From the results, we see that the true values for these propositions are:
- [tex]\( \text{If } x - 5 \neq 10, \text{ then } 4x + 1 \neq 61. \rightarrow False\)[/tex]
- [tex]\( \text{If } 4x + 1 \neq 61, \text{ then } x - 5 \neq 10. \rightarrow False\)[/tex]
- [tex]\( \text{If } x - 5 = 10, \text{ then } 4x + 1 = 61. \rightarrow False\)[/tex]
- [tex]\( \text{If } 4x + 1 = 61, \text{ then } x - 5 = 10. \rightarrow False\)[/tex]
Thus, the inverses all result in false, and none of these options hold true in deriving an inverse for [tex]\( p \rightarrow q \)[/tex].
Let's begin by defining our propositions:
- [tex]\( p \)[/tex]: [tex]\( x - 5 = 10 \)[/tex]
- [tex]\( q \)[/tex]: [tex]\( 4x + 1 = 61 \)[/tex]
First, let's find the values of [tex]\( x \)[/tex] that satisfy [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
1. Solving for [tex]\( p \)[/tex]:
[tex]\[ x - 5 = 10 \][/tex]
[tex]\[ x = 15 \][/tex]
2. Solving for [tex]\( q \)[/tex]:
[tex]\[ 4x + 1 = 61 \][/tex]
[tex]\[ 4x = 60 \][/tex]
[tex]\[ x = 15 \][/tex]
Both propositions are true if [tex]\( x = 15 \)[/tex].
Next, consider the statement [tex]\( p \rightarrow q \)[/tex]. This implication is logically equivalent to [tex]\( \neg p \lor q \)[/tex] (not [tex]\( p \)[/tex] or [tex]\( q \)[/tex]).
Now, let's determine the truth values for different conditions:
1. If [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is true:
From the above solutions, we know that when [tex]\( x = 15 \)[/tex], both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are true:
[tex]\[ p: 15 - 5 = 10 \quad \text{(true)} \][/tex]
[tex]\[ q: 4 \cdot 15 + 1 = 61 \quad \text{(true)} \][/tex]
2. If [tex]\( x - 5 \neq 10 \)[/tex], then [tex]\( 4x + 1 \neq 61 \)[/tex]:
This statement would require that when [tex]\( p \)[/tex] is false ([tex]\( x \neq 15 \)[/tex]), then [tex]\( q \)[/tex] would also be false. However:
- Suppose [tex]\( x = 16 \)[/tex], then [tex]\( x - 5 = 11 \)[/tex] (false for [tex]\( p \)[/tex]), but [tex]\( 4 \cdot 16 + 1 = 65 \)[/tex] (false for [tex]\( q \)[/tex]).
- Suppose [tex]\( x = 14 \)[/tex], then [tex]\( x - 5 = 9 \)[/tex] (false for [tex]\( p \)[/tex]), but [tex]\( 4 \cdot 14 + 1 = 57 \)[/tex] (false for [tex]\( q \)[/tex]).
3. If [tex]\( 4x + 1 \neq 61 \)[/tex], then [tex]\( x - 5 \neq 10 \)[/tex]:
This statement requires that when [tex]\( q \)[/tex] is false, [tex]\( p \)[/tex] must also be false. Again, examining:
- If [tex]\( x = 16 \)[/tex], [tex]\( q \)[/tex] is false ([tex]\( 4 \cdot 16 + 1 = 65 \)[/tex]), and [tex]\( p \)[/tex] is false ([tex]\( x - 5 = 11 \)[/tex]).
- If [tex]\( x = 14 \)[/tex], [tex]\( q \)[/tex] is false ([tex]\( 4 \cdot 14 + 1 = 57 \)[/tex]), and [tex]\( p \)[/tex] is false ([tex]\( x - 5 = 9 \)[/tex]).
4. If [tex]\( x - 5 = 10 \)[/tex], then [tex]\( 4x + 1 = 61 \)[/tex]:
From the values of [tex]\( x = 15 \)[/tex], if [tex]\( p \)[/tex] is true, we see:
[tex]\[ p: 15 - 5 = 10 \quad \text{(true)} \][/tex]
[tex]\[ q: 4 \cdot 15 + 1 = 61 \quad \text{(true)} \][/tex]
5. If [tex]\( 4x + 1 = 61 \)[/tex], then [tex]\( x - 5 = 10 \)[/tex]:
Similarly, if [tex]\( q \)[/tex] is true (with [tex]\( x = 15 \)[/tex]), then:
[tex]\[ q: 4 \cdot 15 + 1 = 61 \quad \text{(true)} \][/tex]
[tex]\[ p: 15 - 5 = 10 \quad \text{(true)} \][/tex]
Given all of these conditions, the inverses of [tex]\( p \rightarrow q \)[/tex] are checked. From the results, we see that the true values for these propositions are:
- [tex]\( \text{If } x - 5 \neq 10, \text{ then } 4x + 1 \neq 61. \rightarrow False\)[/tex]
- [tex]\( \text{If } 4x + 1 \neq 61, \text{ then } x - 5 \neq 10. \rightarrow False\)[/tex]
- [tex]\( \text{If } x - 5 = 10, \text{ then } 4x + 1 = 61. \rightarrow False\)[/tex]
- [tex]\( \text{If } 4x + 1 = 61, \text{ then } x - 5 = 10. \rightarrow False\)[/tex]
Thus, the inverses all result in false, and none of these options hold true in deriving an inverse for [tex]\( p \rightarrow q \)[/tex].
