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Zorluk: KolayForming Contrapositives

Match each conditional statement on the left with its logically valid contrapositive on the right.

  • The treaty will be signed only if both nations compromise.If both nations do not compromise, the treaty will not be signed.
  • Winning the championship requires a strong defense.A team that lacks a strong defense cannot win the championship.
  • Unless the budget is increased, the project will fail.If the project does not fail, the budget was increased.
  • If the alarm sounds, the building must be evacuated and searched.If the building is not evacuated or not searched, the alarm did not sound.

Cevap

The correct matches accurately pair each conditional statement with its logically valid contrapositive by reversing the sufficient and necessary conditions, and properly negating both terms.
Each correct pairing takes the original conditional relationship, successfully reverses the sufficient and necessary conditions, and negates both. It also properly applies De Morgan's laws by swapping 'and' to 'or' where applicable.

Adım Adım Çözüm

1
Identify the sufficient and necessary conditions in each original statement based on logical indicator words (if, only if, requires, unless).
Translated natural language statements into formal conditional logic.
Standardizing the logical structure is a required first step before manipulating the conditions.
2
Apply the rule of contraposition by reversing the order of the conditions and negating both.
Formed the logically equivalent contrapositive for each simple conditional.
The contrapositive is the only formally valid logical deduction that can always be made from a conditional statement.
3
Apply De Morgan's laws to compound conditionals during the negation process.
Converted the 'and' conjunction to an 'or' disjunction when negating the required conditions for the alarm.
When negating a conjunction (A and B), the negation distributes to both terms and changes the operator to a disjunction (~A or ~B).

Anahtar Kavram

Forming Contrapositives
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Konular

Forming Contrapositives

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