Question(a) Given: If Geometry is Maria's favorite subject, then the dress is on sale. If the dress is on sale, then the television is broken. Which statement must also be true? If Geometry is Maria's favorite subject, then the television is broken. (b) Given: If I am in my first period class, then Karen heard the radio. I am in my first period class. Which statement must also be true? I am not in my first period class. (c) Given: If the mouse escaped from the cat, then Yolanda is a doctor. Which statement must also be true? (choose one) (choose one) If Yolanda is not a doctor, then the mouse did not escape from the cat. If Yolanda is a doctor, then the mouse escaped from the cat. If the mouse did not escape from the cat, then Yolanda is not a doctor.
Studdy Solution
STEP 1
1. We are dealing with logical implications and their consequences.
2. We need to determine the logical conclusions based on given premises.
STEP 2
1. Analyze the logical implications in part (a).
2. Analyze the logical implications in part (b).
3. Analyze the logical implications in part (c).
STEP 3
For part (a), we have two implications:
- If Geometry is Maria's favorite subject, then the dress is on sale.
- If the dress is on sale, then the television is broken.
By the transitive property of implications, if the first condition implies the second, and the second implies the third, then the first condition implies the third.
Thus, if Geometry is Maria's favorite subject, then the television is broken.
STEP 4
For part (b), we have:
- If I am in my first period class, then Karen heard the radio.
- I am in my first period class.
This is a direct application of modus ponens, which states that if "If P, then Q" is true and P is true, then Q must be true.
Therefore, Karen heard the radio.
STEP 5
For part (c), we have:
- If the mouse escaped from the cat, then Yolanda is a doctor.
We need to determine the contrapositive, which is logically equivalent to the original statement. The contrapositive of "If P, then Q" is "If not Q, then not P".
Thus, if Yolanda is not a doctor, then the mouse did not escape from the cat.
The conclusions are:
(a) If Geometry is Maria's favorite subject, then the television is broken.
(b) Karen heard the radio.
(c) If Yolanda is not a doctor, then the mouse did not escape from the cat.
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