Consider the following theorem (p -> q):
If the quadrilateral ABCD is a square, then its diagonals are perpendicular and have the
same length.
State the converse (q->p), inverse (~p -> ~q), and contrapositive (~q -> ~p).
"->" is an arrow
well, you have
p: ABCD is a square
q: its diagonals are perpendicular and have the
same length
Now just substitute the words for your logic symbols
To determine the converse, inverse, and contrapositive of the given theorem, we need to understand their definitions.
In a conditional statement of the form "p -> q," where p and q are statements, the following terminology is used:
- Converse: The converse of "p -> q" is "q -> p." It flips the positions of the hypothesis (p) and conclusion (q) of the conditional statement.
- Inverse: The inverse of "p -> q" is "~p -> ~q." It negates both the hypothesis and the conclusion of the conditional statement.
- Contrapositive: The contrapositive of "p -> q" is "~q -> ~p." It flips the positions of the hypothesis and conclusion of the inverse statement.
Now, let's apply these definitions to the given theorem, which is "If the quadrilateral ABCD is a square, then its diagonals are perpendicular and have the same length."
Converse: The converse of the given theorem is "If the diagonals of a quadrilateral are perpendicular and have the same length, then it is a square."
Inverse: The inverse of the given theorem is "If the quadrilateral ABCD is not a square, then its diagonals are not perpendicular or do not have the same length."
Contrapositive: The contrapositive of the given theorem is "If the diagonals of a quadrilateral are not perpendicular or do not have the same length, then it is not a square."
To summarize:
Converse: q -> p
Inverse: ~p -> ~q
Contrapositive: ~q -> ~p