# Difference: EDMExamISampleTest (2 vs. 3)

#### Revision 32011-03-14 - JimSkon

Line: 1 to 1

 META TOPICPARENT name="ElementaryDiscreteMath2011"

# Elementary Discrete Math

###### Sample Exam I
Line: 7 to 7
1. Show whether each of the following is a tautology, contradiction, or neither using truth tables:
Changed:
<
<
(a) p → ¬( p ∨¬q)
>
>
(a) p → ¬( p ∨¬q)
`<--/twistyPlugin twikiMakeVisibleInline-->`
neither
`<--/twistyPlugin-->`

Changed:
<
<
(b) ¬p ∧ ¬q
>
>
(b) ¬p ∧ ¬q
`<--/twistyPlugin twikiMakeVisibleInline-->`
neither
`<--/twistyPlugin-->`

Changed:
<
<
(c) (( p ∨ ¬p) → q)
>
>
(c) (( p ∨ ¬p) → q)
`<--/twistyPlugin twikiMakeVisibleInline-->`
neither
`<--/twistyPlugin-->`

Changed:
<
<
(d) ¬q ∨ (¬qp)
>
>
(d) ¬q ∨ (¬qp)
`<--/twistyPlugin twikiMakeVisibleInline-->`
tautalogy
`<--/twistyPlugin-->`

Changed:
<
<
(e) [( p q ) ∧ ( pr ) ∧ ( qr )]
>
>
(e) [( p q ) ∧ ( pr ) ∧ ( qr )]
`<--/twistyPlugin twikiMakeVisibleInline-->`
neither
`<--/twistyPlugin-->`

Line: 44 to 44
3. Evaluate each of the following bitwise logic expressions
Changed:
<
<
a. (011010 ∧ 111100) ∨ 001100 =
>
>
a. (011010 ∧ 111100) ∨ 001100 =
`<--/twistyPlugin twikiMakeVisibleInline-->`
011000 ∨ 001100 = 011100
`<--/twistyPlugin-->`

Changed:
<
<
b. 111000 ∨ 001100 ∨ 000001 =
>
>
b. 111000 ∨ 001100 ∨ 000001 =
`<--/twistyPlugin twikiMakeVisibleInline-->`
111101
`<--/twistyPlugin-->`

Changed:
<
<
c. (101010 ∨ 110011) → 000111 =
>
>
c. (101010 ∨ 110011) ⊕ 000111 =
`<--/twistyPlugin twikiMakeVisibleInline-->`
111011 ⊕000111 = 111100
`<--/twistyPlugin-->`

4. Prove using truth tables whether each of the following pairs of expressions are equivalent:

Changed:
<
<
a. pq and ( pq) → ( pq)
>
>
a. pq and ( pq) → ( pq)
`<--/twistyPlugin twikiMakeVisibleInline-->`
= yes
`<--/twistyPlugin-->`

Changed:
<
<
b. ( pq) ∨ p and ( pq) ∧ ¬( pq)
>
>
b. ( pq) ∨ p and ( pq) ∧ ¬( pq)
`<--/twistyPlugin twikiMakeVisibleInline-->`
= no
`<--/twistyPlugin-->`

Changed:
<
<
c. ¬( p ∧ ¬q) and pq
>
>
c. ¬( p ∧ ¬q) and pq
`<--/twistyPlugin twikiMakeVisibleInline-->`
= yes
`<--/twistyPlugin-->`

Line: 68 to 68
6. State the converse and contrapositive of each of the following implications:
Changed:
<
<
a. If it snows tonight, then I will stay home
>
>
a. If it snows tonight, then I will stay home.
`<--/twistyPlugin twikiMakeVisibleInline-->`

If I stay home then it will snow tonight.
If I don’t stay home then it will not snow tonight.
`<--/twistyPlugin-->`

Changed:
<
<
b. I go to the beach whenever it is a sunny summer day.

c. When I stay up late, it necessary that I sleep until noon
>
>
b. I go to the beach whenever it is a sunny summer day.
`<--/twistyPlugin twikiMakeVisibleInline-->`

It is a sunny summer day whenever I go to the beach.
It is not a sunny summer day whenever I don’t go to the beach.
`<--/twistyPlugin-->`

c. When I stay up late, it necessary that I sleep until noon
`<--/twistyPlugin twikiMakeVisibleInline-->`

When I sleep until noon, it necessary that I stay up late.

When I don’t sleep until noon, it necessary that I don’t stay up late

`<--/twistyPlugin-->`
Consider the following predicates:
Line: 84 to 88
7. Write the following in symbols:
Changed:
<
<
a. Sandy owns a blue car.
>
>
a. Sandy owns a blue car.
`<--/twistyPlugin twikiMakeVisibleInline-->`
x: C(Sandy,x) ∧ BLUE( x)
`<--/twistyPlugin-->`

Changed:
<
<
b. Every car is either blue or red.
>
>
b. Every car is either blue or red.
`<--/twistyPlugin twikiMakeVisibleInline-->`
∀x: RED( x) ∨ BLUE( x)
`<--/twistyPlugin-->`

Changed:
<
<
c. There is a car that is both blue and red.
>
>
c. There is a car that is both blue and red.
`<--/twistyPlugin twikiMakeVisibleInline-->`
x: BLUE( x) ∧ RED( x)
`<--/twistyPlugin-->`

Changed:
<
<
d. Every blue car has an owner.
>
>
d. Every blue car has an owner.
`<--/twistyPlugin twikiMakeVisibleInline-->`
xy: BLUE( x) → C( y,x)
`<--/twistyPlugin-->`

Changed:
<
<
e. Every red car owner also owns a blue car
>
>
e. Every red car owner also owns a blue car
`<--/twistyPlugin twikiMakeVisibleInline-->`
xyz: (C( x,y) ∧ RED( y)) → (C( x,z) ∧ BLUE( z) )
`<--/twistyPlugin-->`

Line: 111 to 115
8. Tell whether each of the following is true or false, and why.
Changed:
<
<
a. ∃ x : C(Bill, x)
>
>
a. ∃ x : C(Bill, x)
`<--/twistyPlugin twikiMakeVisibleInline-->`
true
`<--/twistyPlugin-->`

Changed:
<
<
b. ∃ x : C(Sandy, x) ∧ BLUE( x)
>
>
b. ∃ x : C(Sandy, x) ∧ BLUE( x)
`<--/twistyPlugin twikiMakeVisibleInline-->`
false (Sandy only owns a red car)
`<--/twistyPlugin-->`

Changed:
<
<
c. ∀x ∃y: C( x,y)
>
>
c. ∀x ∃y: C( x,y)
`<--/twistyPlugin twikiMakeVisibleInline-->`
false (Jack owns no car)
`<--/twistyPlugin-->`

Changed:
<
<
d. ∀y ∃x: C( x,y)
>
>
d. ∀y ∃x: C( x,y)
`<--/twistyPlugin twikiMakeVisibleInline-->`
true
`<--/twistyPlugin-->`

9. Suppose P(x,y) is the statement 2x < y, where x and y are integers. What are the truth values of

Changed:
<
<
a) P(1, -1)
>
>
a) P(1, -1)
`<--/twistyPlugin twikiMakeVisibleInline-->`
false
`<--/twistyPlugin-->`

Changed:
<
<
b) P(0, 0)
>
>
b) P(0, 0)
`<--/twistyPlugin twikiMakeVisibleInline-->`
false
`<--/twistyPlugin-->`

Changed:
<
<
c) ∃y P(3,y)
>
>
c) ∃y P(3,y)
`<--/twistyPlugin twikiMakeVisibleInline-->`
true
`<--/twistyPlugin-->`

Changed:
<
<
d) ∀xy P( x,y)
>
>
d) ∀xy P( x,y)
`<--/twistyPlugin twikiMakeVisibleInline-->`
true
`<--/twistyPlugin-->`

Changed:
<
<
e) ∃xy P( x,y)
>
>
e) ∃xy P( x,y)
`<--/twistyPlugin twikiMakeVisibleInline-->`
false
`<--/twistyPlugin-->`

Changed:
<
<
f) ∀yx P( x,y)
>
>
f) ∀yx P( x,y)
`<--/twistyPlugin twikiMakeVisibleInline-->`
true
`<--/twistyPlugin-->`

Line: 143 to 147
Write each of the following statements using the above predicate and any needed quantifiers:
Changed:
<
<
a) Fran is in room Elmwood 321
>
>
a) Fran is in room Elmwood 321
`<--/twistyPlugin twikiMakeVisibleInline-->`
R(Fran,Elmwood 321)
`<--/twistyPlugin-->`

Changed:
<
<
b) Bill does not have a dorm room.
>
>
b) Bill does not have a dorm room.
`<--/twistyPlugin twikiMakeVisibleInline-->`
¬∃x R(Bill,x)
`<--/twistyPlugin-->`

Changed:
<
<
c) There is a dorm room which is not in use (no assigned students).
>
>
c) There is a dorm room which is not in use (no assigned students).
`<--/twistyPlugin twikiMakeVisibleInline-->`
¬∀yx R( x,y) or ∃yx¬
R( x,y)
`<--/twistyPlugin-->`

Changed:
<
<
d) Every student has a dorm room.
>
>
d) Every student has a dorm room.
`<--/twistyPlugin twikiMakeVisibleInline-->`
xyR( x,y)
`<--/twistyPlugin-->`

Changed:
<
<
e) There is a room with more then one student assigned to it.
>
>
e) There is a room with more then one student assigned to it.
`<--/twistyPlugin twikiMakeVisibleInline-->`
xyzR( x,y) ∧ R( z,y) ∧ x ≠ z
`<--/twistyPlugin-->`

11. Suppose P(x,y) is the statement x + 2y = xy, where x and y are integers. What are the truth values of

Changed:
<
<
(a) P(1,-1)
>
>
(a) P(1,-1)
`<--/twistyPlugin twikiMakeVisibleInline-->`
true
`<--/twistyPlugin-->`

Changed:
<
<
(b) P(0,0)
>
>
(b) P(0,0)
`<--/twistyPlugin twikiMakeVisibleInline-->`
true
`<--/twistyPlugin-->`

Changed:
<
<
(c) ∃yP(3,y)
>
>
(c) ∃yP(3,y)
`<--/twistyPlugin twikiMakeVisibleInline-->`
true
`<--/twistyPlugin-->`

Changed:
<
<
(d) ∀x∃yP( x,y)
>
>
(d) ∀x∃yP( x,y)
`<--/twistyPlugin twikiMakeVisibleInline-->`
true
`<--/twistyPlugin-->`

Changed:
<
<
(e) ∃xyP( x,y)
>
>
(e) ∃xyP( x,y)
`<--/twistyPlugin twikiMakeVisibleInline-->`
False
`<--/twistyPlugin-->`

Changed:
<
<
(f) ∀yxP( x,y)
>
>
(f) ∀yxP( x,y)
`<--/twistyPlugin twikiMakeVisibleInline-->`
False
`<--/twistyPlugin-->`

Changed:
<
<
(g) ∃yxP( x,y)
>
>
(g) ∃yxP( x,y)
`<--/twistyPlugin twikiMakeVisibleInline-->`
False
`<--/twistyPlugin-->`

Changed:
<
<
(h) ¬∀xy ¬P( x,y)
>
>
(h) ¬∀xy ¬P( x,y)
`<--/twistyPlugin twikiMakeVisibleInline-->`
False
`<--/twistyPlugin-->`

12. Find the size (that is, the cardinal number) of each of the following sets:

Changed:
<
<
(a) {x|x ∈ Z and x2 < 10}.
>
>
(a) {x|x ∈ Z and x2 < 10}.
`<--/twistyPlugin twikiMakeVisibleInline-->`
7
`<--/twistyPlugin-->`

Changed:
<
<
(b) P({a,b,c,d}), where P denotes the power set.
>
>
(b) P({a,b,c,d}), where P denotes the power set.
`<--/twistyPlugin twikiMakeVisibleInline-->`
16
`<--/twistyPlugin-->`

Changed:
<
<
(c) {1,3,5,7...}.
>
>
(c) {1,3,5,7...}.
`<--/twistyPlugin twikiMakeVisibleInline-->`
Infinity
`<--/twistyPlugin-->`

Changed:
<
<
(d) A x B where A = {1,2,3,4,5} and B = {1,2,3}.
>
>
(d) A x B where A = {1,2,3,4,5} and B = {1,2,3}.
`<--/twistyPlugin twikiMakeVisibleInline-->`
15
`<--/twistyPlugin-->`

Changed:
<
<
(e) {x|x ∈ N and 9x2 - 1 = 0}.
>
>
(e) {x|x ∈ N and 9x2 - 1 = 0}.
`<--/twistyPlugin twikiMakeVisibleInline-->`
0
`<--/twistyPlugin-->`

Changed:
<
<
(f) P(A), where A is the power set of {a,b,c}.
>
>
(f) P(A), where A is the power set of {a,b,c}.
`<--/twistyPlugin twikiMakeVisibleInline-->`
256
`<--/twistyPlugin-->`

Changed:
<
<
(g) A x B, where A = {a,b,c} and B = Ø.
>
>
(g) A x B, where A = {a,b,c} and B = Ø.
`<--/twistyPlugin twikiMakeVisibleInline-->`
0
`<--/twistyPlugin-->`

Changed:
<
<
(h) {x|x ∈ N and 4x2 - 1 = 0}.
>
>
(h) {x|x ∈ N and 4x2 - 1 = 0}.
`<--/twistyPlugin twikiMakeVisibleInline-->`
0
`<--/twistyPlugin-->`

13. Determine the truth value for each of the following.

Changed:
<
<
(a) ∀x ∃y | (x2=y)

(b) ∀xy | (x=y2)

>
>
(a) ∀x ∃y | (x2=y)
`<--/twistyPlugin twikiMakeVisibleInline-->`
True – every number has a square
`<--/twistyPlugin-->`

(b) ∀xy | (x=y2)
`<--/twistyPlugin twikiMakeVisibleInline-->`
False – No y exists if x is negative

`<--/twistyPlugin-->`

Changed:
<
<
(c) ∃xy | ( xy = 0)
>
>
(c) ∃xy | ( xy = 0)
`<--/twistyPlugin twikiMakeVisibleInline-->`
True – let x = 0

`<--/twistyPlugin-->`

Changed:
<
<
(d) ∃xy | (y ≠ 0 → xy = 1)
>
>
(d) ∃xy | (y ≠ 0 → xy = 1)
`<--/twistyPlugin twikiMakeVisibleInline-->`
False – no number that when multiplied by anything is always 1

`<--/twistyPlugin-->`

Line: 225 to 227
(h) (c,c) ∈ A x A
>
>
`<--/twistyPlugin twikiMakeVisibleInline-->`
(a) F (b) T (c) T (d) T (e) T (f) F (g) F (h) T.
`<--/twistyPlugin-->`

15.Find the size (that is, the cardinal number) of each of the following sets:

Line: 241 to 244
(a) 1, (b) 2^(2^3)=256, (c) ¥ ,(d) 24, (e) 17
>
>
`<--/twistyPlugin twikiMakeVisibleInline-->`
(a) 1, (b) 2^(2^3)=256, (c) infinity ,(d) 24, (e) 17
`<--/twistyPlugin-->`

16. Let A = { 2,4,6,8 } B = {4, 7} C = {Ø, {4, 7}}

Show the following:

Changed:
<
<
a. A ∪ B =
>
>
a. A ∪ B =
`<--/twistyPlugin twikiMakeVisibleInline-->`
{ 2,4,6, 8, 7}
`<--/twistyPlugin-->`

Changed:
<
<
b. A ∩ B =
>
>
b. A ∩ B =
`<--/twistyPlugin twikiMakeVisibleInline-->`
{ 4 }
`<--/twistyPlugin-->`

Changed:
<
<
c. A ∪ C =
>
>
c. A ∪ C =
`<--/twistyPlugin twikiMakeVisibleInline-->`
{{ 2,4,6,8, Ø, {4, 7}}
`<--/twistyPlugin-->`

Changed:
<
<
d. A ∩ C =
>
>
d. A ∩ C =
`<--/twistyPlugin twikiMakeVisibleInline-->`
Ø
`<--/twistyPlugin-->`

Changed:
<
<
e. A - B =
>
>
e. A - B =
`<--/twistyPlugin twikiMakeVisibleInline-->`
{2, 6, 8}
`<--/twistyPlugin-->`

Changed:
<
<
f. C - Ø =
>
>
f. C - Ø =
`<--/twistyPlugin twikiMakeVisibleInline-->`
{Ø, {4, 7}}
`<--/twistyPlugin-->`

Changed:
<
<
g. Ø - C =
>
>
g. Ø - C =
`<--/twistyPlugin twikiMakeVisibleInline-->`
Ø
`<--/twistyPlugin-->`

Changed:
<
<
h. C ∪ (A ∩ B) =
>
>
h. C ∪ (A ∩ B) =
`<--/twistyPlugin twikiMakeVisibleInline-->`
{4, Ø, {4, 7}}
`<--/twistyPlugin-->`

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