Mathematical Logic

The problem references , suggesting a logic circuit with inputs and their negations. Without a specific circuit diagram, assume a common configuration, e.g., an expression involving AND, OR, or XOR gates. A typical circuit might be , which simplifies to XOR.
Let’s hypothesize the output as: Simplify: Conclusion: The circuit likely represents an XOR gate, outputting true when exactly one of or is true.
Answer: Logical expression: . The circuit is equivalent to an XOR gate.

Given statement: If : , then : .
Inverse: If , then : If , then .
Contrapositive: If , then : If , then .
Note: The statement assumes are positive real numbers for to hold, as counterexamples exist (e.g., ).

For to be false, at least one of p or q must be false. For (implication) to be false, p must be true and q must be false (since , false only when p is T and q is F).
Check: If p = T, q = F:
- .
- .
Both conditions are satisfied. Other combinations (e.g., p = F, q = T or p = F, q = F) make true. Thus, p = T, q = F.