Questions: (a) Use one of DeMorgan's Laws to choose a statement equivalent to the following. Walnuts don't grow on lily pads and caterpillars don't turn into butterflies. Equivalent: (Choose one) (b) Use one of DeMorgan's Laws to choose a negation of the following statement. They own a truck or we have a garage. Negation: (Choose one)

Solution Steps

DeMorgan's Laws are fundamental in logic and state the following:

1. The negation of a conjunction (AND) is the disjunction (OR) of the negations: \[ \neg (A \land B) \equiv \neg A \lor \neg B \]
2. The negation of a disjunction (OR) is the conjunction (AND) of the negations: \[ \neg (A \lor B) \equiv \neg A \land \neg B \]

The given statement is: \[ \text{Walnuts don't grow on lily pads and caterpillars don't turn into butterflies.} \] Let:

• \( A = \text{Walnuts grow on lily pads} \)
• \( B = \text{Caterpillars turn into butterflies} \)

The statement can be written as: \[ \neg A \land \neg B \]

Using DeMorgan's Law, the negation of \( \neg A \land \neg B \) is: \[ \neg (\neg A \land \neg B) \equiv A \lor B \]

Thus, the equivalent statement is: \[ \text{Walnuts grow on lily pads or caterpillars turn into butterflies.} \]

The given statement is: \[ \text{They own a truck or we have a garage.} \] Let:

• \( C = \text{They own a truck} \)
• \( D = \text{We have a garage} \)

The statement can be written as: \[ C \lor D \]

Using DeMorgan's Law, the negation of \( C \lor D \) is: \[ \neg (C \lor D) \equiv \neg C \land \neg D \]

Thus, the negation of the statement is: \[ \text{They do not own a truck and we do not have a garage.} \]

Final Answer