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."
Sue writes, "Statement 2 is the converse of statement 3 and contrapositive of statement 1."
Kim writes, "Statement 1 is the inverse of statement 2 and converse of statement 3."
To determine which statements are correct, let's first understand the concepts of converse, inverse, contrapositive, and the negation of a statement:
1. Converse: The converse of a statement switches the order of the original statement's parts. For example, if the original statement is "If A, then B," then the converse would be "If B, then A."
2. Inverse: The inverse of a statement negates both parts of the original statement. For example, if the original statement is "If A, then B," then the inverse would be "If not A, then not B."
3. Contrapositive: The contrapositive of a statement negates and switches the order of the original statement's parts. For example, if the original statement is "If A, then B," then the contrapositive would be "If not B, then not A."
4. Negation: The negation of a statement simply negates the entire statement. For example, if the original statement is "If A, then B," then the negation would be "It is not the case that A implies B."
Now, let's analyze the given statements:
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."
Now, let's check Sue's and Kim's statements:
Sue writes, "Statement 2 is the converse of statement 3 and contrapositive of statement 1."
Kim writes, "Statement 1 is the inverse of statement 2 and converse of statement 3."
Analyzing Sue's statement:
- According to Sue, Statement 2 is the converse of Statement 3. However, this is incorrect. The converse of Statement 3 would be "If she is not stuck in traffic, then she is not late" and is not equivalent to Statement 2.
- According to Sue, Statement 2 is the contrapositive of Statement 1. This is also incorrect. The contrapositive of Statement 1 would be "If she is not late, then she is not stuck in traffic."
Analyzing Kim's statement:
- According to Kim, Statement 1 is the inverse of Statement 2. This is incorrect. The inverse of Statement 2 would be "If she is not late, then she is not stuck in traffic."
- According to Kim, Statement 1 is the converse of Statement 3. This is also incorrect. The converse of Statement 3 would be "If she is not stuck in traffic, then she is not late."
In summary:
- Sue's statement is incorrect.
- Kim's statement is also incorrect.
Therefore, neither Sue's nor Kim's statements are correct.