c318f00t1

c318f00t1 - Now assume statement is true for n. 1/(1*2) +...

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CSc 318 Test #1 Wednesday, 27 September 2000 >>>>>>>SUGGESTED ANSWERS<<<<<<<<<< 1. Prove A 2245 B A × A 2245 B × B. A 2245 B ⇒5 1-1, onto f:A B. Define 1-1, onto g: A × A B × B by g(x,y) = (f(x), f(y)). g is 1-1, because g(x,y)=g(u,v) (f(x),f(y))=(f(u),f(v)) f(x)=f(u) and f(y)=f(v) x=u and y=v (x,y)=(u,v). g is onto, because for (u,v) B × B, u and v B 5 a,b A 220d f(a)=u and f(b)=v g(a,b)=(f(a),f(b))=(u,v). 2. Prove that A B 2 A 2 B . X 2 A X A, but A B, X B X 2 B . 3. Let A={1}, B=A {1,2}. Find B, A × B, B × 2 A , 2 A × B , A-B, |A|, |A-B|, | B × 2 A |, |2 A × B |, 2 |A × B| . B={1, 2}, A × B={(1,1), (1,2)}, B × 2 A ={(1, ), (1,{1}), (2, ),(2,{1})}, 2 A × B ={ , {(1,1)}, {(1,2)}, {(1,1),(1,2)}}, A-B= , |A|=1, |A-B|=0, | B × 2 A |=4, |2 A × B |=4, 2 |A × B| =2 2 =4. 4. Prove that 1/(1*2) + 1/(2*3) + 1/(3*4) + … + 1/(n*(n+1)) = n/(n+1) for all n>0. By induction: Base Case: n=1. 1/(1*2) = 1 / 2 = 1/(1+1).
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Unformatted text preview: Now assume statement is true for n. 1/(1*2) + 1/(2*3) + 1/(3*4) + + 1/((n+1)*(n+2)) = 1/(1*2) + 1/(2*3) + + 1/(n*(n+1)) + 1/((n+1)*(n+2)) = n/(n+1) + 1/((n+1)*(n+2)) = (n 2 + 2n + 1)/(n+1)(n+2)= (n+1)/(n+2) = (n+1)/(n+1+1). 5. Given R = {(1,2), (2,3)} {1, 2, 3} {1, 2, 3}. a. Find the smallest set B such that R B is an equivalence relation on {1, 2, 3}. B={(1,1),(2,2),(3,3),(2,1),(3,2),(1,3),(3,1)} b. Find R o R (the composition of R with R). R o R={(1,3)} c. Find R o R o R. R o R o R = d. Find B such that R B is a function from {1, 2, 3} to {1, 2, 3} which is not 1-1. B = {(3,2)} or B = {(3,3)} e. Find the smallest set B such that R-1 B is an equivalence relation on {1, 2, 3}. B={(1,1),(2,2),(3,3),(1,2),(2,3),(1,3),(3,1)}...
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