Math 150: Discrete Mathematics Midterm 1 2018
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Math 150: Discrete Mathematics
Midterm 1
October 11, 2018
1. (15 points)
(a) Find the conjunction of the propositions p and q where p is the proposition “Jill’s Mac
has more than 16GB free hard disk space” and q is the proposition “The processor in Jill’s Mac runs faster than 1GHz ” .
(b) Let p be the statement “All dogs have flees and are brown” . Write down the negation of p (i.e -p) as an English sentence.
(c) Construct a truth table for the proposition (p A -q) → (p V q).
2. (15 points)
Prove that the following are logically equivalent using truth tables.
(a) p → q = -p ^ q .
(b) -(p n q) = -p ^ -q .
(c) (p → q) ^ (p → r) 三 p → (q ^ r).
3. (15 points)
Translate the following statements into a logical expressions using the standard set notation. (a) “The sum of two positive integers is always positive” .
(b) “For every positive integer, there exists another integer greater than the square root of
the first integer” .
(c) “There do not exist integers such that the sum of their cubes is a perfect cube” .
4. (20 points)
Prove that ^3 is not a rational number.
5. (20 points)
Prove the following identities.
(a) A ∩ B = A u B .
(b) A u B = A ∩ B .
(c) (A) = A.
6. (15 points)
(a) Use the bubble sort algorithm to order the following integers:
2, 4, 6, 5, 1, 7, 10.
(b) Use the binary search algorithm to find the number 7 in the (ordered) list you produce in part (a).
2023-06-02