Statement 1: “If she is stuck in traffic, then she is late.”
Statement 2: “If she is late, then she is stuck in traffic.”
Statement 3: “If she is not late, then she is not stuck in traffic.”
Meg writes, “Statement 3 is the inverse of statement 2 and contrapositive of statement 1.”
Cassandra writes, “Statement 2 is the converse of statement 1 and inverse of statement 3.”
Which option is true?
Answer
A.Only Meg is correct.
B. Only Cassandra is correct.
C. Both Meg and Cassandra are correct.
D. Both Meg and Cassandra are incorrect.
Meg
The answer is D. Both Meg and Cassandra are incorrect.
To determine which option is true, we need to understand the definitions of inverse, converse, contrapositive, and statement substitution.
Inverse: The inverse of a statement switches the original statement and its negation. For example, the inverse of "If she is stuck in traffic, then she is late" is "If she is not stuck in traffic, then she is not late."
Converse: The converse of a statement switches the original statement's hypothesis and conclusion. For example, the converse of "If she is stuck in traffic, then she is late" is "If she is late, then she is stuck in traffic."
Contrapositive: The contrapositive of a statement switches the original statement's hypothesis and conclusion and negates both. For example, the contrapositive of "If she is stuck in traffic, then she is late" is "If she is not late, then she is not stuck in traffic."
Statement Substitution: Statement substitution involves replacing the hypothesis and conclusion in a statement with other statements that have the same truth value.
Let's evaluate each statement made by Meg and Cassandra:
Meg:
- Meg correctly identifies that Statement 3 is the inverse of Statement 2 and the contrapositive of Statement 1.
Cassandra:
- Cassandra states that Statement 2 is the converse of Statement 1 and the inverse of Statement 3. However, this is incorrect. The converse of Statement 1 would be "If she is late, then she is stuck in traffic," and the inverse of Statement 3 would be "If she is late, then she is stuck in traffic."
Based on our analysis, we can conclude that only Meg is correct. Therefore, the correct answer is A. Only Meg is correct.
To determine which option is true, let's first understand the terms being used:
1. Inverse: The inverse of a statement is formed by negating the hypothesis and the conclusion of the original statement.
2. Contrapositive: The contrapositive of a statement is formed by both negating and reversing the hypothesis and conclusion of the original statement.
3. Converse: The converse of a statement is formed by reversing the hypothesis and conclusion of the original statement.
Now let's examine the given statements:
Statement 1: "If she is stuck in traffic, then she is late."
Inverse of Statement 1: "If she is not stuck in traffic, then she is not late."
Statement 2: "If she is late, then she is stuck in traffic."
Converse of Statement 2: "If she is stuck in traffic, then she is late."
Statement 3: "If she is not late, then she is not stuck in traffic."
Contrapositive of Statement 3: "If she is stuck in traffic, then she is late."
Based on the definitions above, we find that Meg is correct in stating that Statement 3 is the inverse of Statement 2 and the contrapositive of Statement 1. Cassandra, on the other hand, is incorrect in stating that Statement 2 is the converse of Statement 1 and the inverse of Statement 3.
Therefore, the correct option is B. Only Meg is correct.