The implication x $\to$ y is equivalent to?

Analyze the question: We need the statement that is logically equivalent to the implication "x → y" (if x, then y).

Concept: Every implication is logically equivalent to its contrapositive. The contrapositive is formed by negating both statements and swapping their positions:

• x → y ≡ ~y → ~x

The other related forms are not equivalent to the original:

• Converse: y → x
• Inverse: ~x → ~y (The converse and inverse are equivalent to each other, but not to the original.)

Truth table check:

| x | y | x → y | ~y → ~x | y → x | ~x → ~y | |---|---|-------|---------|-------|---------| | T | T | T | T | T | T | | T | F | F | F | T | T | | F | T | T | T | F | F | | F | F | T | T | T | T |

The column for x → y (T, F, T, T) matches ~y → ~x (T, F, T, T) exactly. It does not match the converse (y → x) or the inverse (~x → ~y).

Checking the options:

• A. ~y → ~x is the contrapositive, so it is equivalent. ✔
• B. y → ~x is not equivalent.
• C. ~x → ~y is the inverse, so it is not equivalent.
• D. y → x is the converse, so it is not equivalent.

Answer: A

Exam tip: Remember "flip and negate" for the contrapositive. If you flip only (converse) or negate only (inverse), the meaning changes. Only doing both keeps it equivalent.