Taking the contrapositive of this statement?

In summary, the statement can be rewritten as: If every right triangle has an angle defect of zero, then every triangle has an angle defect of zero. The contrapositive of this statement is: If there exists a triangle with a non-zero angle defect, then there exists a non-right triangle with a non-zero angle defect.
  • #1
bonfire09
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Statement: If every right triangle has angle defect equal to zero then the angle defect of every triangle is equal to zero

Taking the contrapositive do i have this correct? : There exists at least one triangle whose angle defect is not zero such that not every right triangle has an angle defect equal to zero.
 
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  • #2
The contrapositive of a conditional is another conditional. But there appears to be no conditional in your version - instead, you've a 'such that'. 'if' at the front and 'then' for 'such that',
 
  • #3
R = the set of all right triangles
T = the set set of triangles
Z = the set of all triangles with angle defect zero

If (for all x in R, x is in Z), then (for all x in T, x is in Z).

The contrapositive of ##p\Rightarrow q## is ##\lnot q\Rightarrow\lnot p##, so the contrapositive of the implication above is

If (there exists an x in T such that x is not in Z), then (there exists an x in R such that x is not in Z).
 

Related to Taking the contrapositive of this statement?

1. What is the contrapositive of a statement?

The contrapositive of a statement is formed by switching the hypothesis and conclusion of the original statement and negating both. For example, if the original statement is "If it rains, then the ground is wet", the contrapositive would be "If the ground is not wet, then it is not raining."

2. Why is taking the contrapositive useful?

Taking the contrapositive of a statement can be useful in logic and mathematical proofs. It allows us to rephrase a statement in a different way that may make it easier to prove or disprove.

3. Can the contrapositive of a statement be true if the original statement is false?

Yes, the contrapositive of a statement can be true even if the original statement is false. This is because the contrapositive is a different statement with a different meaning, and its truth or falsity is not dependent on the original statement.

4. How do you prove the contrapositive of a statement?

To prove the contrapositive of a statement, you must show that if the contrapositive is true, then the original statement must also be true. This can be done through a direct proof, a proof by contradiction, or a proof by contraposition.

5. Are there any statements that are equivalent to their contrapositives?

Yes, there are some statements that are equivalent to their contrapositives. These are known as "self-contrapositives" and include statements of the form "If P, then P" or "If not P, then not P". In these cases, the original statement and its contrapositive are logically equivalent and both are always true.

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