Logical Reasoning: Converse Statement Truth Value

The question asks to identify the statement for which the converse is NOT true. A conditional statement is typically in the form 'If P, then Q' ($P \rightarrow Q$). Its converse is 'If Q, then P' ($Q \rightarrow P$). We need to find the option where the original statement is true (or presented as true), but its converse is false.

Statement Analysis

• Option 1: "If a patient dies even with excellent medical care, he likely had terminal illness."
  • Let P = Patient dies with excellent care.
  • Let Q = Patient had terminal illness.
  • Statement: $P \rightarrow Q$ (qualified by 'likely').
  • Converse: $Q \rightarrow P$ ("If a patient had terminal illness, he likely died even with excellent medical care."). This converse seems plausible or potentially true under the same qualification.
• Option 2: "If a person gets employed, he has good qualifications."
  • Let P = A person gets employed.
  • Let Q = The person has good qualifications.
  • Statement: $P \rightarrow Q$.
  • Converse: $Q \rightarrow P$ ("If a person has good qualifications, he gets employed.").
  • Reasoning: This converse ($Q \rightarrow P$) is NOT necessarily true. A person can possess good qualifications but still face unemployment due to various factors like job market conditions, competition, or lack of opportunities.
• Option 3: "If an integer is even, it is divisible by two."
  • Let P = An integer is even.
  • Let Q = The integer is divisible by two.
  • Statement: $P \rightarrow Q$. This is true by definition.
  • Converse: $Q \rightarrow P$ ("If an integer is divisible by two, it is even."). This is also true by definition.
• Option 4: "If an integer is odd, it is not divisible by two."
  • Let P = An integer is odd.
  • Let Q = The integer is not divisible by two.
  • Statement: $P \rightarrow Q$. This is true by definition.
  • Converse: $Q \rightarrow P$ ("If an integer is not divisible by two, it is odd."). This is also true. Any integer not divisible by two must be odd.

Conclusion

The statement "If a person gets employed, he has good qualifications" (Option 2) is the only one where the converse, "If a person has good qualifications, he gets employed," is demonstrably false. The other statements have true converses.