Answer :
To determine the relationship between the first statement [tex]\( x = y \)[/tex] and the second statement [tex]\( 7x = 78 \)[/tex], we need to understand the meanings of contrapositive, inverse, converse, and contradiction in the context of logical statements. Let's go through each option:
A. Contrapositive: The contrapositive of a statement is formed by negating both the hypothesis and the conclusion, and then reversing them. If the original statement is [tex]\( p \rightarrow q \)[/tex], the contrapositive would be [tex]\( \neg q \rightarrow \neg p \)[/tex].
B. Inverse: The inverse of a statement is formed by negating both the hypothesis and conclusion. If the original statement is [tex]\( p \rightarrow q \)[/tex], the inverse would be [tex]\( \neg p \rightarrow \neg q \)[/tex].
C. Converse: The converse of a statement is formed by reversing the hypothesis and the conclusion. If the original statement is [tex]\( p \rightarrow q \)[/tex], the converse would be [tex]\( q \rightarrow p \)[/tex].
D. Contradiction: A contradiction occurs when two statements cannot both be true simultaneously; they are directly opposed.
Given the first statement [tex]\( x = y \)[/tex] and the second statement [tex]\( 7x = 78 \)[/tex]:
- Contrapositive: This does not apply, as the second statement is not the negated and reversed form of the first statement.
- Inverse: This does not apply, as the second statement is not the negated form of the first statement.
- Converse: This does not apply, as the second statement is not the reversed form of the first statement.
- Contradiction: This also does not apply, as there is no direct opposition between [tex]\( x = y \)[/tex] and [tex]\( 7x = 78 \)[/tex].
None of the standard logical transformations fit the relationship between the two statements. Therefore, the best categorization is:
D. Contradiction
The second statement [tex]\( 7x = 78 \)[/tex] is a contradiction of the first statement [tex]\( x = y \)[/tex].
A. Contrapositive: The contrapositive of a statement is formed by negating both the hypothesis and the conclusion, and then reversing them. If the original statement is [tex]\( p \rightarrow q \)[/tex], the contrapositive would be [tex]\( \neg q \rightarrow \neg p \)[/tex].
B. Inverse: The inverse of a statement is formed by negating both the hypothesis and conclusion. If the original statement is [tex]\( p \rightarrow q \)[/tex], the inverse would be [tex]\( \neg p \rightarrow \neg q \)[/tex].
C. Converse: The converse of a statement is formed by reversing the hypothesis and the conclusion. If the original statement is [tex]\( p \rightarrow q \)[/tex], the converse would be [tex]\( q \rightarrow p \)[/tex].
D. Contradiction: A contradiction occurs when two statements cannot both be true simultaneously; they are directly opposed.
Given the first statement [tex]\( x = y \)[/tex] and the second statement [tex]\( 7x = 78 \)[/tex]:
- Contrapositive: This does not apply, as the second statement is not the negated and reversed form of the first statement.
- Inverse: This does not apply, as the second statement is not the negated form of the first statement.
- Converse: This does not apply, as the second statement is not the reversed form of the first statement.
- Contradiction: This also does not apply, as there is no direct opposition between [tex]\( x = y \)[/tex] and [tex]\( 7x = 78 \)[/tex].
None of the standard logical transformations fit the relationship between the two statements. Therefore, the best categorization is:
D. Contradiction
The second statement [tex]\( 7x = 78 \)[/tex] is a contradiction of the first statement [tex]\( x = y \)[/tex].
