Difference: EDMExamISampleTest (1 vs. 6)

Revision 62011-03-15 - JimSkon

Line: 1 to 1
 
META TOPICPARENT name="ElementaryDiscreteMath2011"

Elementary Discrete Math

Sample Exam I
Line: 39 to 39
 

(h)

Double complement law

( x) = x

(i)

DeMorgan's laws

(x ∧ y) = x ∨ y

(x ∨ y) = x ∧ y
Added:
>
>
(j) Absorbtion (x ∨ y) ∧ x = x
    (x ∧ y) ∨ x = x
 
Line: 185 to 187
  (b) P({a,b,c,d}), where P denotes the power set. ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
16
<--/twistyPlugin-->
Changed:
<
<
(c) {1,3,5,7...}. ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
Infinity
<--/twistyPlugin-->
>
>
(c) {1,3,5,7...}. ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
<--/twistyPlugin-->
  (d) A x B where A = {1,2,3,4,5} and B = {1,2,3}. ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
15
<--/twistyPlugin-->
Line: 276 to 278
  18. Simplify the following using the properties of set operation:
Changed:
<
<
  1. (A ∪ B)c ∩. (A c ∩ B c)
  2. (A ∪ (B ∩ A))
>
>
  1. (A ∪ B)c ∩ (A c ∩ B c) = ... Close
    <--/twistyPlugin twikiMakeVisibleInline-->
    (A ∪ B)c ∩ (A ∪ B )c = (A ∪ B)c = Ac ∪ Bc
    <--/twistyPlugin-->
  2. (A ∪ (B ∩ A)) = ... Close
    <--/twistyPlugin twikiMakeVisibleInline-->
    (A ∪ B) ∩ (A ∪ A ) = (A ∪ B) ∩ (A ) = A
    <--/twistyPlugin-->
 
  1. (A ∪ (B ∩ A)) c ∩ ((C c ∪ B c) ∩ A c) c

Line: 293 to 295
  a. A = {x |x N ∧ (∀y,w : y,w N ∧y ≠ 1 ∧ w ≠ 1 ∧ xyw ) } ( N is the set of natural numbers) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
Prime numbers, infinite
<--/twistyPlugin-->
Changed:
<
<
b. A = {x |x N ∧ x < 100 ∧ (∃y: y N ∧ x = 3y) } ( N is the set of natural numbers) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
multiples of 3 {3, 6, 9, ... 99}, 33
<--/twistyPlugin-->
>
>
b. A = {x |x N ∧ x < 100 ∧ (∃y: y Nx = 3y) } ( N is the set of natural numbers) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
multiples of 3 {3, 6, 9, ... 99}, 33
<--/twistyPlugin-->

c. A = {x |x Z ∧ (∃y: y Zx = y2) } ( Z is the set of Integers) ... Close

<--/twistyPlugin twikiMakeVisibleInline-->
Squares {0, 1, 2, 4, 9, 16, ...}, ∞
<--/twistyPlugin-->
  -- JimSkon - 2011-03-14

Revision 52011-03-15 - JimSkon

Line: 1 to 1
 
META TOPICPARENT name="ElementaryDiscreteMath2011"

Elementary Discrete Math

Sample Exam I
Line: 9 to 9
  (a) p → ( pq) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
neither
<--/twistyPlugin-->
Changed:
<
<
(b) q ∧ ( pq) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
contradiction
<--/twistyPlugin-->
>
>
(b) q ∧ ( pq) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
contradiction
<--/twistyPlugin-->
  (c) (( pp) → q) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
neither
<--/twistyPlugin-->
Line: 193 to 193
  (f) P(A), where A is the power set of {a,b,c}. ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
256
<--/twistyPlugin-->
Changed:
<
<
(g) A x B, where A = {a,b,c} and B = . ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
0
<--/twistyPlugin-->
>
>
(g) A x B, where A = {a,b,c} and B = ∅. ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
0
<--/twistyPlugin-->
  (h) {x|x ∈ N and 4x2 - 1 = 0}. ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
0
<--/twistyPlugin-->
Line: 215 to 215
  (b) {{a}} ⊆ P(A)
Changed:
<
<
(c) ⊆ A
>
>
(c) ∅ ⊆ A
 
Changed:
<
<
(d) {} ⊆ P(A)
>
>
(d) {∅} ⊆ P(A)
 
Changed:
<
<
(e) ⊆ A x A
>
>
(e) ∅ ⊆ A x A
  (f) {a,c} ∈ A
Line: 242 to 242
  (e) {x | x ∈ Z and x2 < 80}.
Changed:
<
<
(a) 1, (b) 2^(2^3)=256, (c) ,(d) 24, (e) 17

... Close

<--/twistyPlugin twikiMakeVisibleInline-->
(a) 1, (b) 2^(2^3)=256, (c) infinity ,(d) 24, (e) 17
<--/twistyPlugin-->
>
>
... Close
<--/twistyPlugin twikiMakeVisibleInline-->
(a) 1, (b) 2^(2^3)=256, (c) ∞ ,(d) 24, (e) 17
<--/twistyPlugin-->
 
Changed:
<
<
16. Let A = { 2,4,6,8 } B = {4, 7} C = {, {4, 7}}
>
>
16. Let A = { 2,4,6,8 } B = {4, 7} C = {∅, {4, 7}}
  Show the following:
Line: 255 to 253
  b. A ∩ B = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
{ 4 }
<--/twistyPlugin-->
Changed:
<
<
c. A ∪ C = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
{{ 2,4,6,8, , {4, 7}}
<--/twistyPlugin-->
>
>
c. A ∪ C = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
{{ 2,4,6,8, ∅, {4, 7}}
<--/twistyPlugin-->
 
Changed:
<
<
d. A ∩ C = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
<--/twistyPlugin-->
>
>
d. A ∩ C = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
<--/twistyPlugin-->
  e. A - B = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
{2, 6, 8}
<--/twistyPlugin-->
Changed:
<
<
f. C - = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
{, {4, 7}}
<--/twistyPlugin-->
>
>
f. C - ∅ = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
{∅, {4, 7}}
<--/twistyPlugin-->

g. ∅ - C = ... Close

<--/twistyPlugin twikiMakeVisibleInline-->
<--/twistyPlugin-->
 
Changed:
<
<
g. - C = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
<--/twistyPlugin-->
>
>
h. C ∪ (A ∩ B) = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
{4, ∅, {4, 7}}
<--/twistyPlugin-->
 
Changed:
<
<
h. C ∪ (A ∩ B) = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
{4, , {4, 7}}
<--/twistyPlugin-->
>
>
i. P(A) = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
{{},{2},{2,4},{2,4,6},{2,4,6,8},{2,4,8},{2,6},{2,6,8},{2,8},{4},{4,6},{4,6,8},{4,8},{6},{6,8},{8}}
<--/twistyPlugin-->
 
Added:
>
>
j. B x C = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
{(4, ∅), (4, {4, 7}), (7, ∅), (7, {4, 7}) }
<--/twistyPlugin-->
 

17. Show Venn Diagrams for each of the above.

Line: 280 to 281
 
  1. (A ∪ (B ∩ A)) c ∩ ((C c ∪ B c) ∩ A c) c

Changed:
<
<
19. Prove the following using a. set builder notation and b Membership tables
>
>
19. Determine which relationship ⊆, =, ⊇ is true between each pair of sets.

a. A ∩ (A ∪ B), A ... Close

<--/twistyPlugin twikiMakeVisibleInline-->
=
<--/twistyPlugin-->

b. A ∩ (B ∪ C), (A ∪ B) ∩ (A ∪ C) ... Close

<--/twistyPlugin twikiMakeVisibleInline-->
<--/twistyPlugin-->

c. A ∪ (B ∩ C), (A ∪ B) ∩ (A ∪ C) ... Close

<--/twistyPlugin twikiMakeVisibleInline-->
=
<--/twistyPlugin-->

20. Consider the following sets. Pick the correct description. What is the cardinality of each set?

 
Changed:
<
<
a. A ∩ (A ∪ B) = A
>
>
a. A = {x |x N ∧ (∀y,w : y,w N ∧y ≠ 1 ∧ w ≠ 1 ∧ xyw ) } ( N is the set of natural numbers) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
Prime numbers, infinite
<--/twistyPlugin-->
 
Changed:
<
<
b. A ∩ (B ∪ C) = (A ∪ B) ∩ (A ∪ C)
>
>
b. A = {x |x N ∧ x < 100 ∧ (∃y: y N ∧ x = 3y) } ( N is the set of natural numbers) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
multiples of 3 {3, 6, 9, ... 99}, 33
<--/twistyPlugin-->
  -- JimSkon - 2011-03-14 \ No newline at end of file

Revision 42011-03-15 - JimSkon

Line: 1 to 1
 
META TOPICPARENT name="ElementaryDiscreteMath2011"

Elementary Discrete Math

Sample Exam I
Line: 9 to 9
  (a) p → ( pq) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
neither
<--/twistyPlugin-->
Changed:
<
<
(b) pq ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
neither
<--/twistyPlugin-->
>
>
(b) q ∧ ( pq) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
contradiction
<--/twistyPlugin-->
  (c) (( pp) → q) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
neither
<--/twistyPlugin-->

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 → ( pq)
>
>
(a) p → ( pq) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
neither
<--/twistyPlugin-->
 
Changed:
<
<
(b) pq
>
>
(b) pq ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
neither
<--/twistyPlugin-->
 
Changed:
<
<
(c) (( pp) → q)
>
>
(c) (( pp) → q) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
neither
<--/twistyPlugin-->
 
Changed:
<
<
(d) q ∨ (qp)
>
>
(d) q ∨ (qp) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
tautalogy
<--/twistyPlugin-->
 
Changed:
<
<
(e) [( p q ) ∧ ( pr ) ∧ ( qr )]
>
>
(e) [( p q ) ∧ ( pr ) ∧ ( qr )] ... Close
<--/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 = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
011000 ∨ 001100 = 011100
<--/twistyPlugin-->
 
Changed:
<
<
b. 111000 ∨ 001100 ∨ 000001 =
>
>
b. 111000 ∨ 001100 ∨ 000001 = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
111101
<--/twistyPlugin-->
 
Changed:
<
<
c. (101010 ∨ 110011) → 000111 =
>
>
c. (101010 ∨ 110011) ⊕ 000111 = ... Close
<--/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) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
= yes
<--/twistyPlugin-->
 
Changed:
<
<
b. ( pq) ∨ p and ( pq) ∧ ( pq)
>
>
b. ( pq) ∨ p and ( pq) ∧ ( pq) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
= no
<--/twistyPlugin-->
 
Changed:
<
<
c. ( pq) and pq
>
>
c. ( pq) and pq ... Close
<--/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. ... Close
<--/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. ... Close
<--/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 ... Close
<--/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. ... Close
<--/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. ... Close
<--/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. ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
x: BLUE( x) ∧ RED( x)
<--/twistyPlugin-->
 
Changed:
<
<
d. Every blue car has an owner.
>
>
d. Every blue car has an owner. ... Close
<--/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 ... Close
<--/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) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
true
<--/twistyPlugin-->
 
Changed:
<
<
b. ∃ x : C(Sandy, x) ∧ BLUE( x)
>
>
b. ∃ x : C(Sandy, x) ∧ BLUE( x) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
false (Sandy only owns a red car)
<--/twistyPlugin-->
 
Changed:
<
<
c. ∀x ∃y: C( x,y)
>
>
c. ∀x ∃y: C( x,y) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
false (Jack owns no car)
<--/twistyPlugin-->
 
Changed:
<
<
d. ∀y ∃x: C( x,y)
>
>
d. ∀y ∃x: C( x,y) ... Close
<--/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) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
false
<--/twistyPlugin-->
 
Changed:
<
<
b) P(0, 0)
>
>
b) P(0, 0) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
false
<--/twistyPlugin-->
 
Changed:
<
<
c) ∃y P(3,y)
>
>
c) ∃y P(3,y) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
true
<--/twistyPlugin-->
 
Changed:
<
<
d) ∀xy P( x,y)
>
>
d) ∀xy P( x,y) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
true
<--/twistyPlugin-->
 
Changed:
<
<
e) ∃xy P( x,y)
>
>
e) ∃xy P( x,y) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
false
<--/twistyPlugin-->
 
Changed:
<
<
f) ∀yx P( x,y)
>
>
f) ∀yx P( x,y) ... Close
<--/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 ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
R(Fran,Elmwood 321)
<--/twistyPlugin-->
 
Changed:
<
<
b) Bill does not have a dorm room.
>
>
b) Bill does not have a dorm room. ... Close
<--/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). ... Close
<--/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. ... Close
<--/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. ... Close
<--/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) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
true
<--/twistyPlugin-->
 
Changed:
<
<
(b) P(0,0)
>
>
(b) P(0,0) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
true
<--/twistyPlugin-->
 
Changed:
<
<
(c) ∃yP(3,y)
>
>
(c) ∃yP(3,y) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
true
<--/twistyPlugin-->
 
Changed:
<
<
(d) ∀x∃yP( x,y)
>
>
(d) ∀x∃yP( x,y) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
true
<--/twistyPlugin-->
 
Changed:
<
<
(e) ∃xyP( x,y)
>
>
(e) ∃xyP( x,y) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
False
<--/twistyPlugin-->
 
Changed:
<
<
(f) ∀yxP( x,y)
>
>
(f) ∀yxP( x,y) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
False
<--/twistyPlugin-->
 
Changed:
<
<
(g) ∃yxP( x,y)
>
>
(g) ∃yxP( x,y) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
False
<--/twistyPlugin-->
 
Changed:
<
<
(h) ∀xy P( x,y)
>
>
(h) ∀xy P( x,y) ... Close
<--/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}. ... Close
<--/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. ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
16
<--/twistyPlugin-->
 
Changed:
<
<
(c) {1,3,5,7...}.
>
>
(c) {1,3,5,7...}. ... Close
<--/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}. ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
15
<--/twistyPlugin-->
 
Changed:
<
<
(e) {x|x ∈ N and 9x2 - 1 = 0}.
>
>
(e) {x|x ∈ N and 9x2 - 1 = 0}. ... Close
<--/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}. ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
256
<--/twistyPlugin-->
 
Changed:
<
<
(g) A x B, where A = {a,b,c} and B = .
>
>
(g) A x B, where A = {a,b,c} and B = . ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
0
<--/twistyPlugin-->
 
Changed:
<
<
(h) {x|x ∈ N and 4x2 - 1 = 0}.
>
>
(h) {x|x ∈ N and 4x2 - 1 = 0}. ... Close
<--/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) ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
True – every number has a square
<--/twistyPlugin-->


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

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

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

<--/twistyPlugin-->
 
Line: 225 to 227
  (h) (c,c) ∈ A x A
Added:
>
>
... Close
<--/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
Added:
>
>
... Close
<--/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 = {, {4, 7}}

Show the following:

Changed:
<
<
a. A ∪ B =
>
>
a. A ∪ B = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
{ 2,4,6, 8, 7}
<--/twistyPlugin-->
 
Changed:
<
<
b. A ∩ B =
>
>
b. A ∩ B = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
{ 4 }
<--/twistyPlugin-->
 
Changed:
<
<
c. A ∪ C =
>
>
c. A ∪ C = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
{{ 2,4,6,8, , {4, 7}}
<--/twistyPlugin-->
 
Changed:
<
<
d. A ∩ C =
>
>
d. A ∩ C = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
<--/twistyPlugin-->
 
Changed:
<
<
e. A - B =
>
>
e. A - B = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
{2, 6, 8}
<--/twistyPlugin-->
 
Changed:
<
<
f. C - =
>
>
f. C - = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
{, {4, 7}}
<--/twistyPlugin-->
 
Changed:
<
<
g. - C =
>
>
g. - C = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
<--/twistyPlugin-->
 
Changed:
<
<
h. C ∪ (A ∩ B) =
>
>
h. C ∪ (A ∩ B) = ... Close
<--/twistyPlugin twikiMakeVisibleInline-->
{4, , {4, 7}}
<--/twistyPlugin-->
 

Revision 22011-03-14 - JimSkon

Line: 1 to 1
 
META TOPICPARENT name="ElementaryDiscreteMath2011"

Elementary Discrete Math

Sample Exam I
Added:
>
>
Exam Date: March 28, 2011
 1. Show whether each of the following is a tautology, contradiction, or neither using truth tables:

(a) p → ( pq)

Line: 34 to 36
 

x ∨ x = x

(g)

Complement laws

x ∧ x = F

x ∨ x = T

Changed:
<
<

(h)

Double complement law

(x) = x

(i)

DeMorgan's laws

(x ∧ y) = x ∨ y

(x ∨ y) = x ∧ y
>
>

(h)

Double complement law

(x) = x

(i)

DeMorgan's laws

(x ∧ y) = x ∨ y

(x ∨ y) = x ∧ y
 

Revision 12011-03-14 - JimSkon

Line: 1 to 1
Added:
>
>
META TOPICPARENT name="ElementaryDiscreteMath2011"

Elementary Discrete Math

Sample Exam I

1. Show whether each of the following is a tautology, contradiction, or neither using truth tables:

(a) p → ( pq)

(b) pq

(c) (( pp) → q)

(d) q ∨ (qp)

(e) [( p q ) ∧ ( pr ) ∧ ( qr )]


2. Show whether each of the expression in problem 1 is a tautology, contradiction, or neither using the properties of logic. (logical equivalence's):

Logical Equivalences Table

(a)

Associative laws

x(yz) = (xy)z

x ∨ (y ∨ z) = (x ∨ y) ∨ z

(b)

Commutative laws

x ∧ y = y ∧ x

x ∨ y = y ∨ x

(c)

Distributive laws

x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)

x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)

(d)

Identity laws

x ∧ T = x

x ∨ F = x

(e)

Dominance laws

x ∧ F = F

x ∨ T = T

(f)

Idempotent laws

x ∧ x = x

x ∨ x = x

(g)

Complement laws

x ∧ x = F

x ∨ x = T

(h)

Double complement law

(x) = x

(i)

DeMorgan's laws

(x ∧ y) = x ∨ y

(x ∨ y) = x ∧ y


3. Evaluate each of the following bitwise logic expressions

a. (011010 ∧ 111100) ∨ 001100 =

b. 111000 ∨ 001100 ∨ 000001 =

c. (101010 ∨ 110011) → 000111 =


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

a. pq and ( pq) → ( pq)

b. ( pq) ∨ p and ( pq) ∧ ( pq)

c. ( pq) and pq


5. Show whether each of the expression in problem 4 are equivalent using the properties of logic. (logical equivalence's):


6. State the converse and contrapositive of each of the following implications:

a. If it snows tonight, then I will stay home

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

Consider the following predicates:

C( x,y) - true if person x owns car y.

RED( x) - true if x is a red car.

BLUE( x) - true if x is a blue car.


7. Write the following in symbols:

a. Sandy owns a blue car.

b. Every car is either blue or red.

c. There is a car that is both blue and red.

d. Every blue car has an owner.

e. Every red car owner also owns a blue car


Now consider a world for the above predicates with only 4 people and four cars:

a. Bill owns a blue car and a red car.

b. Sandy owns a red car.

c. Robin owns a blue car.

d. Jack does not own a car.

For the following, consider ONLY the data above:


8. Tell whether each of the following is true or false, and why.

a. ∃ x : C(Bill, x)

b. ∃ x : C(Sandy, x) ∧ BLUE( x)

c. ∀x ∃y: C( x,y)

d. ∀y ∃x: C( x,y)


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

a) P(1, -1)

b) P(0, 0)

c) ∃y P(3,y)

d) ∀xy P( x,y)

e) ∃xy P( x,y)

f) ∀yx P( x,y)


10. Suppose the variable x represents students and y represents dorm rooms:

R( x,y): student x is in dorm room y.

Write each of the following statements using the above predicate and any needed quantifiers:

a) Fran is in room Elmwood 321

b) Bill does not have a dorm room.

c) There is a dorm room which is not in use (no assigned students).

d) Every student has a dorm room.

e) There is a room with more then one student assigned to it.


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

(a) P(1,-1)

(b) P(0,0)

(c) ∃yP(3,y)

(d) ∀x∃yP( x,y)

(e) ∃xyP( x,y)

(f) ∀yxP( x,y)

(g) ∃yxP( x,y)

(h) ∀xy P( x,y)


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

(a) {x|x ∈ Z and x2 < 10}.

(b) P({a,b,c,d}), where P denotes the power set.

(c) {1,3,5,7...}.

(d) A x B where A = {1,2,3,4,5} and B = {1,2,3}.

(e) {x|x ∈ N and 9x2 - 1 = 0}.

(f) P(A), where A is the power set of {a,b,c}.

(g) A x B, where A = {a,b,c} and B = .

(h) {x|x ∈ N and 4x2 - 1 = 0}.


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

(a) ∀x ∃y | (x2=y)


(b) ∀xy | (x=y2)



(c) ∃xy | ( xy = 0)

(d) ∃xy | (y ≠ 0 → xy = 1)


14. Suppose A = {a,b,c}. Mark each of the following TRUE or FALSE:

(a) {b,c} ⊆ P(A)

(b) {{a}} ⊆ P(A)

(c) ⊆ A

(d) {} ⊆ P(A)

(e) ⊆ A x A

(f) {a,c} ∈ A

(g) {a,b} ∈ A x A

(h) (c,c) ∈ A x A


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

(a) {x |x ∈ Z and x2 = 2x}

(b) P(A), where A = P(P({1,2,3})

(c) {1, 3, 5, 7...}

(d) S x T, where S = {a,b,c,d} and T = {1,2,3,4,5,6}

(e) {x | x ∈ Z and x2 < 80}.

(a) 1, (b) 2^(2^3)=256, (c) ,(d) 24, (e) 17


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

Show the following:

a. A ∪ B =

b. A ∩ B =

c. A ∪ C =

d. A ∩ C =

e. A - B =

f. C - =

g. - C =

h. C ∪ (A ∩ B) =


17. Show Venn Diagrams for each of the above.


18. Simplify the following using the properties of set operation:

  1. (A ∪ B)c ∩. (A c ∩ B c)
  2. (A ∪ (B ∩ A))
  3. (A ∪ (B ∩ A)) c ∩ ((C c ∪ B c) ∩ A c) c

19. Prove the following using a. set builder notation and b Membership tables

a. A ∩ (A ∪ B) = A

b. A ∩ (B ∪ C) = (A ∪ B) ∩ (A ∪ C)

-- JimSkon - 2011-03-14

 
This site is powered by the TWiki collaboration platformCopyright &© by the contributing authors. All material on this collaboration platform is the property of the contributing authors.
Ideas, requests, problems regarding TWiki? Send feedback