Prove the following statement using an element argument and reasoning directly from the definitions of union, intersection, set difference.
For all sets A, B, and C, (A ? B) ? C ? A ? (B ? C).
Proof that (A [ B) \ C (A [ (B \ C):
Suppose that x is any element in (A ? B) ? C. [We must show that x ? A ? (B ? C).]
By definition of intersection, x ? (A ? B) and x ? C.
And, by definition of union, x ? A or x ? B, and in both cases, x ? C.
Case 1 (x 2 A and x 2 C): Then, by definition of intersection, x ? B ? C, and so by definition of union, x ? A ? (B ? C).
Case 2 (x 2 B and x =2
B): Then, by definition of set difference, x ? B ? C, and so by
definition of union, x ? A ? (B ? C).
In both cases, x ? A ? (B ? C) [as was to be shown].
So every element in (A ? B) ? C is in A ? (B ? C), and hence (A ? B) ? C ? A ? (B ? C) by
definition of subset.
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Reverse the order of integration and then evaluate the integral.
A. 
(e - 1)
B. 
(e - 1)
C. 
(2e - 1)
D. 
(2e - 1)
Solve.To best determine the fuel economy of a particular car model, an automotive technician calculates the mean gas mileage for 20 full tanks of gas.(a) Use classes of 1.0 mi/gal, beginning with 19.0 - 19.9 mi/gal, to compute the mean fuel economy (in miles per gallon) from the following data:20.419.622.520.522.119.820.520.219.020.619.821.719.921.520.719.621.419.319.421.5(b) At $3.39 per gallon, use the mean fuel economy to calculate the average cost per mile to operate this vehicle.
A. (a) 20.14 mpg, (b) $0.168 (or 16.8¢ per mile) B. (a) 21.05 mpg, (b) $0.161 (or 16.1¢ per mile) C. (a) 20.45 mpg, (b) $0.166 (or 16.6¢ per mile) D. (a) 21.75 mpg, (b) $0.156 (or 15.6¢ per mile)
If possible, factor the polynomial completely. If a polynomial cannot be factored, state that it is prime.64k3m - 144k2m2 + 81km3
A. km(8k - 9m)(8k + 9m) B. km(8k + 9m)2 C. (64k2m + km)(k + 81m2) D. km(8k - 9m)2
Graph.36(x - 1)2 - 16(y - 2)2 = 576
A. 
B. 
C. 
D. 