Sample Second Midterm Key

# Sample Second Midterm Key - Sample Second Midterm Key MATH...

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Sample Second Midterm Key MATH 108 Second Midterm Sample Key 1. a. A relation from a set A to a set B is any subset of the Cartesian product of A and B. b. Given a set A, a subset of the universe U, the characteristic function of A is the function from U to {0,1 } given by mapping the elements of A to 1 and the elements of U-A to 0. c. A surjection is an onto function. That is, it is a function whose range is the same set as its codomain. 2. (<=) (a,b,c) = ((a,b), c) = ((x,y), z) = (x,y,z) 1/2 QED (=>) (a,b,c) = (x,y,z) So, ((a,b), c) = ((x,y), z) But this means that c = z. Also, (a,b) = (x,y) But this means that a = x and b = y. QED 3. a. R is not an equivalence relation, since it is not reflexive. To see this let x elt Q. Then x = 3^k x => k=0, but 0 not elt N. [It is also not symmetric, as xRy => x = 3^k y => y = 3^-k x, but -k not elt N. It is transitive -- see next.] b. Since 0 elt Z, R is reflexive by the argument above. x = 3^k y => y = 3^-k x, and hence R is symmetric. If xRy and yRz, then x = 3^k y and y = 3^j z. Then x = 3^k (3^j z) = 3^(k+j) z, and hence R is transitive. Thus R is an equivalence relation. 1/2 / R = {y elt Q | 1/2 R y} = {y elt Q | 1/2 = 3^ky} = { . . . , 1/18, 1/6, 1/2, 3/2, . . . } 4. a. Clearly phi is a relation on phi cross phi, as the empty set is a subset of any set, including phi cross phi.

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Sample Second Midterm Key - Sample Second Midterm Key MATH...

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