Let be the set {,,} and be the set {,}.

a. Is a subset of ? b. Is a subset of ? c. What is?

d. What is? e. What is? f. What is the power set of ?

If set has elements and set has elements, how many elements are in ? Explain your answer.

If set has elements, how many elements are in the power set of ? Explain your answer.

Find the error in the following proof that 2 = 1: Consider the equation = . Multiply both sides by to obtain ! = . Subtract ! frombothsidestoget! ! =!.Nowfactoreachside,(+)()= ( ), and divide each side by ( ) to get + = . Finally, let equal 1, which shows that 2 = 1.

Let()=1+2++bethesumofthefirstnaturalnumbersandlet()= 1" + 2" + + " be the sum of the first cubes. Prove the following equalities by induction on , to arrive at the curious conclusion that () = !() for every .

a. ()=#(+1) !

b. ()=#($ +2" +!)=#!(+1)!

1. Let A be the set {x,y,z} and B be the set {x,y}. a. Is A a subset of B ? b. Is B a subset of A ? c. What is AB ? d. What is AB ? e. What is AB ? f. What is the power set of B ? 2. If set A has a elements and set B has b elements, how many elements are in AB ? Explain your answer. 3. If set C has c elements, how many elements are in the power set of C ? Explain your answer. 4. Find the error in the following proof that 2=1 : Consider the equation a=b. Multiply both sides by a to obtain a2=ab. Subtract b2 from both sides to get a2b2=abb2. Now factor each side, (a+b)(ab)= b(ab), and divide each side by (ab) to get a+b=b. Finally, let a and b equal 1 , which shows that 2=1. 5. Let S(n)=1+2++n be the sum of the first n natural numbers and let C(n)= 13+23++n3 be the sum of the first n cubes. Prove the following equalities by induction on n, to arrive at the curious conclusion that C(n)=S2(n) for every n. a. S(n)=21n(n+1) b. C(n)=41(n4+2n3+n2)=41n2(n+1)2