CS173 Discrete Mathematical Structures
Fall 2006
Homework #3 Solutions
Homework #3 was graded on a scale of 39 points.
1. 10 points total. 2 points for each part: 2 for correct, 1 for incorrect with some
explanation, 0 for incorrect with no argument.
a. False
b. False
c. True
d. True
e. True
2. 6 points total. 2 points for each part: 1 point for the correct power set and 1 point
for the correct cardinality.
a. {∅}, cardinality 1
b. {∅, {∅}}, cardinality 2
c. {∅, {∅}, {{∅}}, {∅, {∅}}}, cardinality 4
3. 12 points total. 4 points for each part: 4 for the correct response and no errors in
justification, 3 for a correct response but minor errors in justification, 1 for
incorrect response with an attempt at justification, 0 for no attempt.
a. 1. (A ∩ B) ⊆ A Given
2. ∀x (x ∈ (A ∩ B) → x ∈ A) Definition of Set Membership
3. ∀x ((x ∈ A ∧ x ∈ B) → x ∈ A) Definition of Intersection
4. ∀x (¬ (x ∈ A ∧ x ∈ B) ∨ x ∈ A) Implication
5. ∀x (x ∉ A ∨ x ∉ B ∨ x ∈ A) DeMorgan
6. ∀x ((x ∉ A ∨ x ∈ A) ∨ x ∉ B) Associativity
7. ∀x (T ∨ x ∉ B) Tautology
8. ∀x (T) Domination
b. 1. A ⊆ A ∪ B Given
2. ∀x (x ∈ A → x ∈ (A ∪ B)) Definition of Set Membership
3. ∀x (x ∈ A → (x ∈ A ∨ x ∈B)) Definition of Union
4. ∀x (¬ (x ∈ A) ∨ (x ∈ A ∨ x ∈B)) Implication
5. ∀x ((x ∉ A ∨ x ∈ A) ∨ x ∈B)) Negation, Associativity
6. ∀x (T ∨ x ∈B)) Tautology
7. ∀x (T) Domination
