Verified Answer :

  • Assign propositional variables \(p\) for "x=4" and \(q\) for "y=-2".
  • Recognize the statement "If x=4, then y=-2" as a logical implication of the form "If p, then q".
  • Identify that the symbol \(\rightarrow\) represents the conditional implication in propositional logic.
  • Select the option that uses this operator, which is \(p \rightarrow q\), corresponding to option (b).

Final Answer: \(\boxed{b}\)

Explanation

  1. Analyzing the Logical Structure

We are presented with the statement "If \(x=4\), then \(y=-2\)" and are asked to identify its symbolic representation from a list of options. In propositional logic, a conditional statement follows the structure "If P, then Q," where P is the antecedent (the condition) and Q is the consequent (the result). Let \(p\) represent the proposition "x=4" and \(q\) represent the proposition "y=-2". Our statement is therefore "If \(p\), then \(q\)."

  1. Evaluating Logical Operators

Now, let's examine the meaning of the given logical operators to match our statement:

  • \(p \wedge q\) represents the conjunction, which is translated as "p AND q". This describes a situation where both events happen simultaneously.
  • \(p \rightarrow q\) represents the conditional (or implication), which is translated as "IF p THEN q". This perfectly captures the conditional relationship given in our statement.
  • \(p \vee q\) represents the disjunction, which is translated as "p OR q". This implies at least one of the conditions is true.
  • \(p \leftrightarrow q\) represents the biconditional, which is translated as "p IF AND ONLY IF q". This indicates that the truth values of \(p\) and \(q\) are equivalent.
  1. Determining the Correct Symbolic Representation

Comparing our structure "If \(p\), then \(q\)" with the definitions of the operators, it is clear that the implication operator \(p \rightarrow q\) is the correct representation. Therefore, the symbolic form of "If \(x=4\), then \(y=-2\)" is \(p \rightarrow q\). This corresponds to option (b).

Examples

In computer programming, logical conditions are essential for decision-making. For example, consider an algorithm that processes a sensor signal. A programmer might write: "If the temperature is greater than 100 degrees, then trigger the alarm." This is an example of an implication \(p \rightarrow q\), where \(p\) is "temperature > 100" and \(q\) is "trigger alarm". This logical structure allows computers to perform automated tasks based on specific, predefined rules.

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