In the questions below P(m n) means “m ≤ n”, where the ...
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MATH/EECS 1019 (Fall 2016)
Test 1
Instructions:
• The exam is 80 minutes long
• You cannot use books, notes, cell phones or any other materials
• Please write your answers next to the questions and not on a separate sheet of paper (you can use additional sheets for your own calculations)
In the questions below determine whether the proposition is TRUE or FALSE
1. 1 + 1 ’ 3 if and only if 2 + 2 ’ 3.
Ans: True
2. If 1 < 0, then 3 ’ 4.
Ans: True
3. If 1 + 1 ’ 2 or 1 + 1 ’ 3, then 2 + 2 ’ 3 and 2 + 2 ’ 4.
Ans: False
4. Write the truth table for the proposition ¬(r → ¬q) ∨ (p ∧ ¬r).
|p | q | r | ¬(r → ¬q) ∨ (p ∧ ¬r) |
|Ans:| | | |
|p | | | |
|T |T |T |T |
|T |T |F |T |
|T |F |T |F |
|T |F |F |T |
|F |T |T |T |
|F |T |F |F |
|F |F |T |F |
|F |F |F |F |
| | | | |
5. Determine whether this proposition is a tautology: ((p → ¬q) ∧ q) → ¬p.
Ans: Yes. (a truth table should be presented with all T in the last column)
In the questions below P(x,y) means “x + 2y ’ xy”, where x and y are integers. Determine the truth value of the statement.
6. P(1,−1).
Ans: True
7. P(0,0).
Ans: True
8. ∃yP(3,y).
Ans: True
9. ∀x∃yP(x,y).
Ans: False
10. ∃x∀yP(x,y).
Ans: False
11. ∀y∃xP(x,y).
Ans: False
12. ∃y∀xP(x,y).
Ans: False
13. ¬∀x∃y¬P(x,y).
Ans: False
In the questions below suppose the variable x represents people, and
F(x): x is friendly T(x): x is tall A(x): x is angry.
Write the statement using these predicates and any needed quantifiers.
14. Some people are not angry.
Ans: ∃x¬A(x).
15. All tall people are friendly.
Ans: ∀x(T(x) → F(x)).
16. No friendly people are angry.
Ans: ∀x(F(x) → ¬A(x)).
17. Give a proof by contradiction of the following: “If n is an odd integer, then n2 is odd”.
Ans: Suppose n ’ 2k + 1 but n2 ’ 2l. Therefore (2k + 1)2 ’ 2l, or 4k2 + 4k + 1 ’ 2l. Hence 2(2k2 + 2k − l) ’ −1 (even = odd), a contradiction. Therefore n2 is odd.
18. Consider the following theorem: If n is an even integer, then n + 1 is odd. Give a proof by contraposition of this theorem.
Ans: Suppose n + 1 is even. Therefore n + 1 ’ 2k. Therefore n ’ 2k − 1 ’ 2(k − 1) + 1, which is odd.
19. Give a proof by cases that [pic]for all real numbers x.
Ans: Case 1, [pic]: then [pic], so [pic]. Case 2, [pic]: here [pic] and [pic], so [pic].
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.
20. {∅,{a}}.
Ans: Yes, {a}.
21. {∅,{a},{∅,a}}.
Ans: No, it lacks {∅}.
22. {∅,{a},{∅},{a,∅}}.
Ans: Yes, {{a,∅}}.
23. {∅,{a,∅}}.
Ans: No, it lacks {a} and {∅}.
24. Find [pic]([pic]).
Ans: ([pic])
In the questions below, suppose A ’ {a,b,c} and B ’ {b,{c}}. Mark the statement TRUE or FALSE
25. c ∈ A − B.
Ans: True
26. | P(A × B) | ’ 64.
Ans: True
27. ∅ ∈ P(B).
Ans: True
28. {c} ⊆ B.
Ans: False
29. {a,b} ∈ A × A.
Ans: False
30. {b,c} ∈ P(A).
Ans: True
31. {b,{c}} ∈ P(B).
Ans: True
32. ∅ ⊆ A × A.
Ans: True
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