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Onto Functions - Discrete Structures - Solved Exam, Exams of Discrete Structures and Graph Theory

English and Foreign Languages UniversityDiscrete Structures and Graph Theory

These solved exam paper are very easy to understand and very helpful to built a concept about the foundation of computers and discrete structures.The key points discuss in these notes are:Onto Functions, Tautology, Contradiction, Map Sets, Negation of Implication, Density of Binary String, Conditional Statement, Set of Rational Numbers, Laws of Logic, Universal Conditional, Rules of Inference

Typology: Exams

2012/2013

Uploaded on 04/27/2013

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Spring 2011 Examination 1 CMSC 203 Name SOLUTION KEY
1. (20 points) Circle T for True or F for False as they apply to the following statements:
T F A statement is either a tautology or a contradiction.
T F The set {a, e, i, o, u, y} has 64 subsets.
T F The empty set is a subset of itself.
T F Onto functions map sets to sets of equal or smaller size.
T F If Σ
= {0, 1} , then Σ
3
= {000, 111}.
T F The negation of an implication is an implication.
T F The density of a binary string equals its length.
T F If Σ
= {0, 1}, then for any string s in Σ
8
, H(s, 11111111) = d( s ).
T F The conditional statement and its converse are logically equivalent.
T F The set of Rational numbers is countable.
2. (10 points) Use the Laws of Logic to show:
¬
p
∧
(q
∨
¬
r)
≡
¬
[ (r
→
q)
→
p ]
¬
p
∧
(q
∨
¬
r)
≡
¬
p
∧
(
¬
r
∨
q)
≡
¬
p
∧
(r
→
q)
≡
¬
[ p
∨
¬
(r
→
q) ]
≡
¬
[
¬
(r
→
q)
∨
p ]
≡
¬
[ (r
→
q)
→
p ]
3. (6 points) Find the negation of the following Universal Conditional:
Some people who like Math study Economics.
All people like Math and do not study Economics.
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1. (20 points) Circle T for True or F for False as they apply to the following statements:

T F A statement is either a tautology or a contradiction.

T F The set {a, e, i, o, u, y} has 64 subsets.

T F The empty set is a subset of itself.

T F Onto functions map sets to sets of equal or smaller size.

T F If Σ = {0, 1} , then Σ^3 = {000, 111}.

T F The negation of an implication is an implication.

T F The density of a binary string equals its length.

T F If Σ = {0, 1}, then for any string s in Σ^8 , H( s , 11111111) = d( s ).

T F The conditional statement and its converse are logically equivalent.

T F The set of Rational numbers is countable.

2. (10 points) Use the Laws of Logic to show: ¬ p ∧ ( q ∨ ¬ r ) ≡ ¬ [ ( r → q ) → p ]

¬ p ∧ ( q ∨ ¬ r ) ≡ ¬ p ∧ (¬ r ∨ q ) ≡ ¬ p ∧ ( r → q ) ≡ ¬ [ p ∨ ¬ ( r → q ) ]

≡ ¬ [ ¬ ( r → q ) ∨ p ]

≡ ¬ [ ( r → q ) → p ]

3. (6 points) Find the negation of the following Universal Conditional: Some people who like Math study Economics.

All people like Math and do not study Economics.

4. (10 points) Use the Rules of Inference to show the following is a valid argument:

p → q

¬ r → ¬ q

r → ( s ∧ u )

p

∴ s

1. p → q AND p THEREFORE q ;

2. ¬ r → ¬ q AND q THEREFORE r ;

3. r → ( s ∧ u ) AND r THEREFORE ( s ∧ u );

4. ( s ∧ u ) THEREFORE s.

5. (10 points) Given the alphabet Σ = {0,1}, list the subsets of Σ^4 where each subset contains the

elements of Σ^4 that have the same Hamming Distance from 1100.

s 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 H(1100, s ) 2 3 3 4 1 2 2 3 1 2 2 3 0 1 1 2

H(1100, s ) = 0: { 1100 } H(1100, s ) = 1: { 0100, 1000, 1101, 1110 } H(1100, s ) = 2: { 0000, 0101, 0110, 1001, 1010, 1111 } H(1100, s ) = 3: { 0001, 0010, 0111, 1011 } H(1100, s ) = 4: { 0011 }

6. (10 points) Using the Properties of Sets, to show (A − B) ∪ (A ∩ B) ∪ (B − A) = A ∪ B.

(A − B) ∪ (A ∩ B) ∪ (B − A) = (A ∩ Bc) ∪ (A ∩ B) ∪ (B ∩ Ac)

= (A ∩ Bc) ∪ (A ∩ B) ∪ (A ∩ B) ∪ (B ∩ Ac) = [ A ∩ (Bc^ ∪ B)] ∪ [(A ∪ Ac) ∩ B] = (A ∩ U) ∪ (U ∩ B) = A ∪ B.