Subjects logic

At Least One Wrong

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At Least One Wrong


1. The phrase "at least one is wrong" typically refers to a situation where among multiple statements or conditions, one or more are incorrect. 2. To analyze this mathematically, consider a set of statements $S = \{s_1, s_2, \ldots, s_n\}$. 3. "At least one is wrong" means the negation of "all are correct". 4. If we let $C_i$ represent the correctness of statement $s_i$, then "all are correct" is $\bigwedge_{i=1}^n C_i$. 5. Therefore, "at least one is wrong" is $\neg \left( \bigwedge_{i=1}^n C_i \right)$. 6. By De Morgan's laws, this is equivalent to $\bigvee_{i=1}^n \neg C_i$, meaning at least one $C_i$ is false. 7. This logical expression helps in reasoning about correctness in problems involving multiple conditions or statements. Final answer: "At least one is wrong" means $\bigvee_{i=1}^n \neg C_i$, i.e., at least one statement is false.