91Ó°ÊÓ

Consider the statement about a party, "If it's your birthday or there will be cake, then there will be cake." (a) Translate the above statement into symbols. Clearly state which statement is \(P\) and which is \(Q\). (b) Make a truth table for the statement. (c) Assuming the statement is true, what (if anything) can you conclude if there will be cake? (d) Assuming the statement is true, what (if anything) can you conclude if there will not be cake? (e) Suppose you found out that the statement was a lie. What can you conclude?

Consider the statement "for all integers \(a\) and \(b\), if \(a+b\) is even, then \(a\) and \(b\) are even" (a) Write the contrapositive of the statement. (b) Write the converse of the statement. (c) Write the negation of the statement. (d) Is the original statement true or false? Prove your answer. (e) Is the contrapositive of the original statement true or false? Prove your answer. (f) Is the converse of the original statement true or false? Prove your answer. (g) Is the negation of the original statement true or false? Prove your answer.

Consider the statement: for all integers \(n,\) if \(n\) is even then \(8 n\) is even. (a) Prove the statement. What sort of proof are you using? (b) Is the converse true? Prove or disprove.

The game TENZI comes with 40 six-sided dice (each numbered 1 to 6 ). Suppose you roll all 40 dice. (a) Prove that there will be at least seven dice that land on the same number. (b) How many dice would you have to roll before you were guaranteed that some four of them would all match or all be different? Prove your answer.

Write the negation, converse and contrapositive for each of the statements below. (a) If the power goes off, then the food will spoil. (b) If the door is closed, then the light is off. (c) \(\forall x\left(x<1 \rightarrow x^{2}<1\right)\) (d) For all natural numbers \(n,\) if \(n\) is prime, then \(n\) is solitary. (e) For all functions \(f,\) if \(f\) is differentiable, then \(f\) is continuous. (f) For all integers \(a\) and \(b\), if \(a \cdot b\) is even, then \(a\) and \(b\) are even. (g) For every integer \(x\) and every integer \(y\) there is an integer \(n\) such that if \(x>0\) then \(n x>y\) (h) For all real numbers \(x\) and \(y\), if \(x y=0\) then \(x=0\) or \(y=0\). (i) For every student in Math 228 , if they do not understand implications, then they will fail the exam.

Prove that for all integers \(n,\) it is the case that \(n\) is even if and only if \(3 n\) is even. That is, prove both implications: if \(n\) is even, then \(3 n\) is even, and if \(3 n\) is even, then \(n\) is even.

Consider the statement: for all integers \(a\) and \(b\), if \(a\) is even and \(b\) is a multiple of 3 , then \(a b\) is a multiple of 6 . (a) Prove the statement. What sort of proof are you using? (b) State the converse. Is it true? Prove or disprove.

Consider the statement: for all integers \(n,\) if \(n\) is odd, then \(7 n\) is odd. (a) Prove the statement. What sort of proof are you using? (b) Prove the converse. What sort of proof are you using?

Suppose you have a collection of 5 -cent stamps and 8 -cent stamps. We saw earlier that it is possible to make any amount of postage greater than 27 cents using combinations of both these types of stamps. But, let's ask some other questions: (a) Prove that if you only use an even number of both types of stamps, the amount of postage you make must be even. (b) Suppose you made an even amount of postage. Prove that you used an even number of at least one of the types of stamps. (c) Suppose you made exactly 72 cents of postage. Prove that you used at least 6 of one type of stamp.

You come across four trolls playing bridge. They declare: Troll 1: All trolls here see at least one knave. Troll 2: I see at least one troll that sees only knaves. Troll 3: Some trolls are scared of goats. Troll 4: All trolls are scared of goats. Are there any trolls that are not scared of goats? Recall, of course, that all trolls are either knights (who always tell the truth) or knaves (who always lie).