91Ó°ÊÓ

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can't use Euler diagrams to determine the validity of an argument if one of the premises is false.

Give an example of a conditional statement that is true, but whose converse and inverse are not necessarily true. Try to make the statement somewhat different from the conditional statements that you have encountered throughout this section. Explain why the converse and the inverse that you wrote are not necessarily true.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The inverse of a statement's converse is the statement's contrapositive.

From Alice in Wonderland: "This time she found a little bottle and tied around the neck of the bottle was a paper label, with the words DRINK ME beautifully printed on it in large letters. It was all very well to say DRINK ME, but the wise little Alice was not going to do that in a hurry. 'No, I'll look first,' she said, 'and see whether it's marked poison or not,' for she had never forgotten that if you drink much from a bottle marked poison, it is almost certain to disagree with you, sooner or later. However, this bottle was not marked poison, so Alice ventured to taste it." Alice's argument: If the bottle is marked poison, I should not drink from it. This bottle is not marked poison. \(\therefore\) I should drink from it. Translate this argument into symbolic form and determine whether it is valid or invalid.

From Alice in Wonderland: "Alice noticed, with some surprise, that the pebbles were all turning into little cakes as they lay on the floor, and a bright idea came into her head. 'If I eat one of these cakes,' she thought, 'it's sure to make some change in my size; and as it can't possibly make me larger, it must make me smaller, I suppose." " Alice's argument: If I eat the cake, it will make me larger or smaller. It can't make me larger. \(\therefore\) If I eat the cake, it will make me smaller. Translate this argument into symbolic form and determine whether it is valid or invalid.

Give an example of a disjunction that is true, even though one of its component statements is false. Then write the negation of the disjunction and explain why the negation is false.

Let \(p, q\), and \(r\) represent the following simple statements: \(p\) : The temperature outside is freezing. \(q\) : The heater is working. \(r\) : The house is cold. Write each compound statement in symbolic form. Sufficient conditions for the house being cold are freezing outside temperatures and a heater not working.

Explain how to form the negation of a given English statement. Give an example.

Explain how to write the negation of a quantified statement in the form "All \(A\) are \(B\)." Give an example.

Determine the truth value for each statement when \(p\) is false, \(q\) is true, and \(r\) is false. \(\sim p \leftrightarrow(\sim q \wedge r)\)