1.1/1.2 1a
During a local campaign, eight Republican and five Democratic candidates are nominated for president of the school board. a. If the president is to be one of these candidates, how many possibilities are there for the eventual winner?
8 + 5 = 13

1.4 1a
In how many ways can 10 (identical) dimes be distributed among five children if (a) there are no restrictions? The number of combinations of n objects taken r at a time is: C ( n + r – 1 , r ) x1 + x2 + x3 + ... + xn = r x1 + x2 + x3 + x4 + x5 = 10
Translates to: number of dimes given to person 1
+ number of dimes given to person 2
+ number of dimes given to person 3
+ number of dimes given to person 4
+ number of dimes given to person 5 10 total dimes n = 5 r = 10
C ( 5 + 10 – 1 , 10 ) C ( 14 , 10 ) 1,001

2.1 2
Identify the primitive statements in problem 1: a. In 2003 George W. Bush was the president of the United States. b. x+3 is a positive integer. c. Fifteen is an even number. d. If Jennifer is late for the party, then her cousin Zachary will be quite angry. e. What time is it? f. As of June 30, 2003, Christine Marie Evert had won the French open a record seven times.

2.2 2
Verify the Absorption Law by means of a truth table. p ∨ (p ∧ q)<->p p or (p and q) p | q | p q | p (p q) | p ∨ (p ∧ q)<->p | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

2.3 2b
Use truth tables to verify that each of the following is a logical implication: b. [[(p ∧ q)→r] ∧ ￢q ∧ (p→￢r)]→(￢p ∨￢q) p | q | r | p∧q | [(p∧q)→r] | [(p∧q)→r]∧￢q | (p→￢r) | [[(p∧q)→r]∧￢q∧(p→￢r)] | (￢p ∨￢q) | [[(p ∧ q)→r] ∧ ￢q ∧ (p→￢r)] →(￢p ∨￢q) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 |

2.4 1
Let p(x), q(x) denote the following open statements. p(x): x<3 q(x): x+1 is odd
If the universe consists of all integers, what are the truth values of the following statements? a. q(1) a. False b. ¬p(3) b. False c. p(7) ∨ q(7) c. False d. p(3) ∧ q(4) d. True e. ￢(p(−4) ∨ q(−3)) e. False f. ￢p(−4)∧￢q(−3) f. False

2.5 1
In example 2.52 (below) why did we stop at 26 and not 28?
Suppose we start with the universe that comprises only the 13 integers 2, 4, 6, 8, ..., 24, 26. Then we can establish the statement:
For all n (meaning n = 2, 4, 6, 8, ..., 26), we can write n as the sum of at most 3 perfect squares.
The results in Table 2.24 provide a case-by-case verification showing the given (quantified) statement to be true. (We call this statement a theorem).
Table 2.24 2=1+1 | 10=9+1 | 20=16+4 | 4=4 | 12=4+4+4 | 22=9+9+4 | 6=4+1+1 | 14=9+4+1 | 24=16+4+4 | 8=4+4 | 16=16 | 26=25+1 | | 18=16+1+1 | | For example 2.52 we stopped at 26 because 28 can be written as the sum of more than 3 perfect squares, as shown below. Therefore, we cannot write 28 as the sum of AT MOST 3 perfect squares.
28=9+9+9+1
28=25+1+1+1
28=16+9+1+1+1

3.1 1
Which of the following sets are equal? a. {1, 2, 3} b. {3, 2, 1, 3} c. {3, 1, 2, 3} d. {1, 2, 2, 3}
They are all the same set because order and repetition do not matter so long as the sets contain the same elements.

3.2 3a
Determine the sets A, B where
A − B = {1, 3, 7, 11},
B − A = {2, 6, 8}, and
A ∩ B = {4, 9}. A = {1, 3, 7, 11} B = {2, 6, 8}
4, 9
4, 9 A ∩ B = {4, 9} A={1, 3, 4, 7, 9, 11}
B={2, 4, 6, 8, 9}

3.3 1
During freshman orientation at a small liberal arts college, two showings of the latest James Bond movie were presented. Among the 600 freshmen, 80 attended the first showing and 125 attended the second showing, while 450 didn’t make it to either showing. How many of the 600 freshmen attended twice? First Showing = a
Second Showing = b
Both Showings = c
Neither Showing = d a+b+c+d=600 a+c=80 b+c=125 d=450

a+b+c+450=600 a+125+450=600 a=600-450-125 a=25 25+c=80 c=80-25 c=55

b+c=125 b+55=125 b=125-55 b=70 Since c=55 and c was both showings, we know that 55 students attended twice.-450=150 1}} he 600 freshmen mall liberal arts college, s the sum of more than 3 perfect squares, as shown below.