contrapositive of p→q
The contrapositive of the statement “p implies q” is a logically equivalent statement that is derived by switching the hypothesis and conclusion of the original statement and negating both
The contrapositive of the statement “p implies q” is a logically equivalent statement that is derived by switching the hypothesis and conclusion of the original statement and negating both.
In symbols, the contrapositive of “p→q” can be represented as “¬q→¬p” or “not q implies not p.”
For example, consider the original statement: “If it is raining (p), then the ground is wet (q).”
The contrapositive of this statement would be: “If the ground is not wet (¬q), then it is not raining (¬p).”
To see the logical equivalence of the contrapositive, we can examine truth tables. Here is the truth table for the original statement “p→q” and its contrapositive “¬q→¬p”:
| p | q | p→q | ¬q | ¬p | ¬q→¬p |
|—|—|——|—-|—-|——-|
| T | T | T | F | F | T |
| T | F | F | T | F | T |
| F | T | T | F | T | F |
| F | F | T | T | T | T |
As you can see from the truth table, the values for “p→q” and “¬q→¬p” are the same in all cases, indicating that the contrapositive is logically equivalent to the original statement.
The contrapositive is often used in mathematics for proving statements or arguments. It can be a helpful technique because sometimes it is easier to work with the contrapositive statement rather than the original one.
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