please answer anything you know

Dad .Dad, I learned today that if I study hard then I will be able to end racism.Dad, today I learned about (AB) C and (AB) C

This got this great student to ask a question that reminded me of our dynamic. Even Jeff got muddled about it at first. I just dont understand my dad.Maybe if I do well at school, he will love me. Maybe if I follow all his rules, he will love me. I know that at least one of these is true. I just dont know which.

1.I hope that this does not sound familiar. In your own intuition, what does she need to do to guarantee his love?

2.Denote A I do well at school. B I follow all his rules. C He will love me. X Her situation. Express X in terms of A, B, and C. Consider (A\veeB) \vee ightarrowC (A\wedgeB) 1 C Hint: One of these is equivalent to X. Let X denote that one. And Z the other.

3.The girl misinterpreted her dynamic with her father as Z. Believing in Z, what strategy would she follow?Did this work out for her? Why?

4. Because X and Z are not equivalent, there must be a universe/model/assignment of T/F to A, B, & Cso that one of X and Z is true and the other is false. Give such an assignment and evaluate X and Z under it.

5.Prove XX Consider the top operator in X, i.e., , , Use a rule in the left-hand-using column of rules in the purple table. Do not use Contrapositive or De Morgan's Law.

6-Prove X'X. The standard algorithm for tautologies is called Davis-Putnam. The ideal is to try setting a variable (in this case C) to true and then using this information simplifying the formula. Then it does the same when setting C to false. sam turns this idea into a proof system. Your proof should follow this idea by use cases C and C Twice we are now going to prove that X and X are equivalent. We will use three fun techniques.

7. Prove that the dual of X the dual of X. Duality is a technique we will be learning much later. Good and Evil are considered Duals of each other: Though different, they can be the flip side of the same coin. They paired. The dual of the dual of A is A. Understanding one gives a deeper understanding of the other.

7. Prove that the dual of X the dual of X. Duality is a technique we will be learning much later. Take the dual of X and of X as follows: Take the contra positive of each . This produces negations. Use De Morgan's Law to bubble the negations in till each gets to the variables. Replace A, B, and C with A, B, and C. Use your own words to argue that the dual of Xand that of X are equivalent.