### Question Description

Exam: 9/30 at 1:30-2:00 pm EST

There will be 5 questions about negate quantified statements. Translate English into Math, negate and translate it back. Direct proof and proof by Contrapositive. Be able to provide contrapositive of given implications.

I will ask you for help me answer the questions that I don’t know.

Here’s the homework examples for this class.

Problem 1: Indicate whether each statement is true or false. For each false statement, explainwhy the statement is false.(a) ? ? ?(b) ? ? {?}(c) ? = {?}(d) ? ? {?}(e) ? ? ?(f) ? ? {?}(g) {3, 7, 4} = {4, 3, 7}

Problem 2: For statements P and Q, construct a truth table for (P =? Q) =? (? P)

Problem 3: State the negation of each of the following sentences. Be careful!(a) At least two of my socks are missing.(b) One of my two sisters is an opera singer.(c) No one expected that aliens live among us.(d) Its surprising that Alireza Firouzja did not qualify for the Candidates Tournament.

Problem 4: For statements P, Q, R show the following is a tautology.((P ? Q) ? (Q ? R)) ? (P ? R)

Problem 5: For statements P and Q, which of the following implies P ? Q is false?(a) (? P) ? (? Q) is false.(b) (? P) ? Q is true.(c) (? P) ? (? Q) is true.(d) Q ? P is true.(e) P ? Q is false.

Problem 6: Let P be the statement There exist integers a and b such that ab < 0 anda + b > 0.(a) Express P symbolically.(b) Determine ? P as a positive statement.(b) Re-express ? P in words.

Problem 7: Do the same as in problem 6 for the statement For all real numbers x and y,x 6= y implies that x2 + y2 > 0.Problem 8: For statements P and Q, show that((P ? Q)? ? (P ? Q)) ? ? (P ?? Q)

Problem 9: Four friends are discussing going to a dance recital. Aida says shell go if Baolegoes. Baole says hell go if Carmen goes. Carmen says shell go if Domingo goes. Exactly twoof the friends then went to the recital. Assuming everyone told the truth, which two went?Analyze this problem logically. Set up four logical statements of the form A : Aida goes tothe dance recital. Restate each of the friends statements as an implication. Then construct atruth table for the set of implications. The truth table should allow you to deduce which of thefriends went to the dance recital.

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