solve number 2 Show, using induction: For n N, Prove: GROUP B Prove: If A B then

Question

solve number 2

Show, using induction: For n N, Prove: GROUP B Prove: If A B then Prove: If A B then A (BA) = B. GROUP C Prove (perhaps using the Pigeonhole principle): Show that a simple connected graph with at least two vertices must have two vertices with the identical degree. Prove (perhaps with the contrapositive): If 5x - y is even then x + y is even. Suppose for a,b,n,d,k,l,m Z, d divides a,b and n. That is, a = kd, b = ld and n = md. Show that if a b( mod n) then k l( mod m). GROUP D Prove the following: If 6000 Americans attend a Mathematics conference, at least 120 of them must be from the same state. Suppose that a rally is attended by 8000 people. Show that at least ten of them have the same first and last initials in their name.

Explanation / Answer

by induction method

for n = 1 we have

1 = 2-1 =1

assuming for it for n and solving for n+1

summ j*j! = (n+1)! -1 --> assumed

for n+1

summ j*j! + (n+1)* (n+1)! = (n+2)! -1

summ j*j!= (n+2)* (n+1)! -(n+1)* (n+1)!-1

= (n+1)! (n+2-n-1)-1= (n+1)! -1

hence proved by induction

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