Discrete Mathematics Answers

At the beginning of the first day (day 1) after grape harvesting is completed, a grape grower has 8000 kg of grapes in storage. On day n, for n = 1, 2, . . . , the grape grower sells 250n/(n + 1) kg of the grapes at the local market at the price of $2.50 per kg. He leaves the rest of the grapes in storage where each day they dry out a little so that their weight decreases by 3%. Let wn be the weight (in kg) of the stored grapes at the beginning of day n for n ≥ 1 (before he takes any to the market).

(a) Find the value of wn for n = 2.

(b) Find a recursive definition for wn. (You may find it helpful to draw a timeline.)

(c) Let rn be the total revenue (in dollars) earned from the stored grapes from the beginning of day 1 up to the beginning of day n for n ≥ 1. Find a recursive formula for rn.

Find, showing all working, a recursive denition of the sequence with general

term

tn = 3 (n + 1)!/2n ; n >= 1

What is the difference between an outcome and an event?

show that p↔q and (p∧q)∨(¬p∧¬q) are logically equivalent

Show that ¬(¬p) and p are logically equivalent.

Choose the best answer: ANTI SYMMETRIC, TRANSITIVE RELATION, SYMMETRIC RELATION, REFLEXIVE RELATION

1. {(3,5)(5,3) (2,4)(4,5)}

2. {(3,1)(1,3)}

3. {(3,1)(2,3)(5,6)(6,5)}

4. {(1,3)(5,3)(7,5)}

5. {(8,9)(9,7)(8,7)}

6. {(4,2)(2,4)}

7. {(1,5)(2,5)(3,5)}

8. {(5,5)(5,6)(6,5)(6,6)(6,7)(7,6) (7,7)}

9. {(6,5)(5,4)(6,4)}

10. {(7,6)(6,5)(7,5)}

11. {(3,1)(1,3)}

12. {(4,3)(3,5)(4,5)}

13. {(6,6)(6,5)(5,5)(5,4)(4,4)}

14. {(6,5)(7,6)(4,5)(5,4)}

15. {(3,3)(4,4)(4,5)(5,4)(5,5)}

A survey of 100 college students revealed the following information about their enrolment in Mathematics, Political Science, and Sociology courses. Twenty-six take Mathematics; 65 take Political Science; 65 take Sociology; 14 take Mathematics and Political Science; 13 take Mathematics and Sociology; 40 take Political Science and Sociology; 8 take Mathematics, Political Science and Sociology. How many students are taking both Mathematics and Sociology but not Political Science?