Key Concepts and Formulas

• Conditional Statement: A conditional statement "If p, then q" is denoted by $p \to q$.
• Negation of a Conditional Statement: The negation of $p \to q$ is given by $p \land \sim q$.
• Logical Connectives:
  • $\sim$ denotes negation (NOT)
  • $\land$ denotes conjunction (AND)
  • $\lor$ denotes disjunction (OR)
  • $\to$ denotes implication (IF...THEN)

Step-by-Step Solution

Step 1: Define the propositions.

Let's define the following propositions: $p$: I become a teacher. $q$: I will open a school.

We are given the statement "If I become a teacher, then I will open a school", which can be represented as $p \to q$.

Step 2: Find the negation of the conditional statement.

We want to find the negation of the statement $p \to q$, which is denoted by $\sim (p \to q)$. Using the formula for the negation of a conditional statement, we have: $\sim (p \to q) \equiv p \land \sim q$ This means "p AND NOT q".

Step 3: Translate the negation back into English.

Now, we translate $p \land \sim q$ back into English using the definitions of $p$ and $q$: $p$: I become a teacher. $\sim q$: I will not open a school.

Therefore, $p \land \sim q$ translates to "I will become a teacher and I will not open a school."

Common Mistakes & Tips

• A common mistake is to confuse the negation of $p \to q$ with $\sim p \to \sim q$ or $\sim p \lor q$. Remember that $\sim (p \to q) \equiv p \land \sim q$.
• Another mistake is to incorrectly translate the logical connectives back into English. Pay close attention to the meaning of "and" and "or".
• Remember the equivalent form $p \to q \equiv \sim p \lor q$. Therefore, $\sim(p \to q) \equiv \sim(\sim p \lor q) \equiv p \land \sim q$ (using De Morgan's Law).

Summary

We are given the statement "If I become a teacher, then I will open a school", which is represented as $p \to q$. We need to find the negation of this statement, $\sim (p \to q)$. Using the formula $\sim (p \to q) \equiv p \land \sim q$, we find the negation to be "I will become a teacher and I will not open a school." This corresponds to option (A).

Final Answer The final answer is \boxed{I will become a teacher and I will not open a school}, which corresponds to option (A).