Discrete Mathematics
Examination II
Key

1. Find the midpoint of the interval [-10, 20].   Answer: 5

2. Find the point ¾ of the way from 5 to 20.   Answer: 65/4

3. If 8 is 3/5 of the way from 2 to b, what is b?  Answer: b = 12

4. Let A = {1,2,3,4,6,7,8,9,10,11,12,13,14,15,16}.  Give { x | x Î A and 4 < x <= 14 } Answer: {6, 7, 8, 9, 10, 11, 12, 13, 14}

5. Give the power set of {x, y, z}  Answer:  { {}, {x}, {y}, {z}, {x,y}, {x,z}, {y,z}, {x,y,z} }

6. The set X contains 25 elements.  How many elements are in the power set of 2X?  Answer:  225

7.   Let A = {Ottawa, Raleigh, Durham, Nashville}
      Let B = {Ottawa, Pomona, Nashville, Salem, Frankfort}

a. Find A È B  Answer: { Ottawa, Raleigh, Durham, Nashville, Pomona, Salem, Frankfort }
b. Find A Ç B  Answer: { Ottawa, Nashville }

c. Find B - A  Answer:  { Pomona, Salem, Frankfort }

8. Name the five components of a mathematical system.
Answer:  primative terms, axioms, definitions, conjectures, theorems.
9. Let A = {a, b, c, d} and B={b, c, d, e}.  Give a Venn diagram representing A and B.  Within that Venn diagram indicate where the following values should go: a, b, c, d, e, f, g, h.  Answer:  requires graphics.  See Dr. Crawford

10. Direct proofs

a. Describe the concept of a direct proof.
    Answer:
        To prove: If P then Q
            1.  Assume P
            2.  Use definitions, axioms, and theorems to arrive at Q.
                    This will be done using syllogisms
b. Prove, using a direct proof:  If x is divisible by 5 then x2 is divisible by 25.
Answer
Assume:  x is divisible by 5 (using the definition of divisible)
Then x = 5p for some integer p
Thus x2 = 25p2
Thus x2 is divisible by 25.  (using the definition of divisible)
11. Give the truth table for p ® q

Answer

P    Q    P ® Q

T    T      T
T    F      F
F    T      T
F    F      F

12. Give the truth table for p Ú q
Answer
P    Q    P ÚQ

T    T      T
T    F      T
F    T      T
F    F      F

13.  Give the truth table for p Ù ~q.  How does this compare with p ® q?
Answer
P    Q    ~Q    P Ù ~Q

T    T     F      F
T    F     T      T
F    T     F      F
F    F     T      F

p Ù ~q   and  p ® q  have opposite truth tables.  They are negations of each other.

14. Consider the following statement: If 9 divides x2 then 9 divides x.
a. Give the hypothesis of that statement.
Answer:  9 divides x2
b. Give the conclusion of that statement.
Answer:  9 divides x
c. Give the contrapositive of that statement.
Answer:  if  9 does not divide x then  9 does not divide x2