Download ¬p∨q. Contrapositive: If you do not sleep late, then it is not Saturday

CSE 240 Exam 1
Spring 2016
Name:___________________________
1. (2 pts) Write a proposition equivalent to p → q using only p, q, ¬, and the connective ∨.
¬p∨q.
2. (3 pts) Write the contrapositive, converse, and inverse of the following: You sleep late if it is
Saturday.
Contrapositive: If you do not sleep late, then it is not Saturday. Converse: If you sleep
late, then it is Saturday. Inverse: If it is not Saturday, then you do not sleep late.
3. (3 pts) Find three subsets of {1,2,3,4,5,6,7,8,9} such that the intersection of any two has size 2
and the intersection of all three has size 1.
For example,{1,2,3},{2,3,4},{1,3,4}.
4.(2pts) Find a proposition using only p, q, ¬, and the connective ∨with the truth table below.
P
T
T
F
F
Q
T
F
T
F
¬(p∨¬q).
?
F
F
T
F
2
In the questions below determine whether the given set is the power set of some set. If the set is a
power set, give the set of which it is a power set. (1 pt each)
5. {Ø, {Ø}, {a}, {{a}}, {{{a}}}, {Ø, a}, {Ø, {a}}, {Ø, {{a}}}, {a, {a}}, {a, {{a}}},
{{a}, {{a}}}, {Ø,a,{a}}, {Ø,a,{{a}}}, {Ø,{a},{{a}}}, {a,{a},{{a}}}, {Ø,a,{a},{{a}}}}.
Ans: Yes {Ø,a,{a},{{a}}}.
6.{∅,{∅,a}}
No, it lacks {a} and {Ø}.
In questions 7 and 8 below suppose A={a,b,c}. Mark the statement TRUE or FALSE.
(1 pt each)
7. {b,c}∈P(A)
Ans: True
8. {a,c}∈A
Ans: FALSE;
In 9 –12 P(x,y) means “x+2y = xy”, where x and y are integers. Determine whether the statement
is true or false. (1 pt each)
9. P(1, −1) - True
10. P(0, 0) - True
11. ∃y P(3,y) – True
12. ∀x∃y P(x, y). – False
3
In 13 - 14 P(x, y) means “x and y are real numbers such that x + 2y = 5”. Determine whether the
statement is true or false. (1 pt each)
13. ∀x∃y P(x, y).
True
14. ∃x∀y P(x, y). False
15. (2pts) Determine whether the following argument is valid. Name the rule of inference or the
fallacy.
If n is a real number such that n>2, then n2 >4. Suppose that n≤2. Then n2 ≤4.
Not valid: fallacy of denying the hypothesis.
In 16 - 17 suppose the variables x and y represent real numbers, and
L(x,y) : x < y
G(x) : x > 0
P(x) : x is a prime number.
Write the statement in good English without using any variables in your answer. (1 pt each)
16. ∀x∃y L(x, y)
There is no largest number.
17. ∀x∃y [G(x) → (P (y) ∧ L(x, y))]
No matter what positive number is chosen, there is a larger prime.
Use the following to answer questions 17-21: (1 pt each)
In the questions below suppose the variable x represents students, F(x) means “x is a freshman”,
and M(x) means “x is a math major”. Match the statement in symbols with one of the English
statements in this list:
1. Some freshmen are math majors.
2. Every math major is a freshman.
3. No math major is a freshman.
18. ∀x(M(x)→F(x))
( 2)
19. ∃x(F(x)∧M(x)) ( 1)
20. ¬∀x (¬F (x) ∨ ¬M (x)) = ∃x ¬(¬ F (x) ∨ ¬M (x)) = ∃x(F(x)∧M(x)) (1)
21. ∀x (¬(M (x) ∧ ¬F (x)) = ∀x (¬M (x) ∨ F (x)) = ∀x(M(x)→F(x)) (2)
22. ∀x (¬M (x) ∨ ¬F (x)) = ¬∃x(F(x)∧M(x)) (3)
4
23. (4 pts) Consider the following theorem: If n is an even integer, then n + 1 is odd. Give a
proof by contraposition of this theorem.
Suppose n+1 is even. Therefore n+1=2k. Therefore n=2k−1 =2(k−1)+1, which is
odd.
24. (4 pts) Prove the following theorem: n is even if and only if n2 is even.
If n is even, then n2 = (2k)2 = 2(2k2), which is even. If n is odd, then n2 = (2k+1)2 =
2(2k2 +2k)+1, which is odd.