COMP9020-Set 1 Numbers, Sets, Words and Logic Solved

1.  (Numbers)

How many numbers in the interval [1431,9758] are

a.  divisible by 3?

b.  divisible by 5?

c.  divisible by 3 and 5?

d.  divisible by 3 or 5?

2.  (Sets)

Prove that (A ∖ B) ∪ (B ∖ A) = (A ∪ B) ∖ (A ∩ B)

a.  using Venn diagrams,

b.  without Venn diagrams.

[hide answer]

3.  (Alphabets and Words)

Let Σ = {a,b,c} and Ψ = {a,c,e}.

a.  How many words are in the set Σ2 ?

b.  What are the elements of Σ2 ∖ Ψ∗ ?

c.  Is it true that Σ∗ ∖ Ψ∗ = (Σ ∖ Ψ)∗ ?  Why or why not?

(Propositional Logic)

For each of the following formulae, determine all the truth assignments to A, B and C under which the formula is true.

a.  A ∧ (¬C ⇒ (B ∨ ¬A))

b.  (A ∧ ¬C) ⇒ (B ∨ ¬A)

c.  (¬C ⇒ ¬A) ∧ (B ⇒ (A ∧ ¬C))

d.  ¬(C ⇒ A) ∧ (A ∨ (B ∧ ¬C))

4.  (Proving properties of algorithms)

Recall the algorithm for computing the greatest common divisor (gcd) of two positive numbers:

⎧⎪ m

gcd(m,n) = ⎨⎩⎪ gcd(m − n,n)

gcd(m,n − m)
if m = n if m n if m < n
Recall the correctness proof given in class. What needs to be changed to adapt it to the faster version below?

m     if n = 0 gcd(n,mmodn)         if n 0

gcd(m,n) = {

6. Challenge Exercise
A multiplication magic square has the product of the numbers in each row, column and diagonal the same. If the diagram below is filled with positive integers to form a multiplicative magic square, then give the value of y.

 1.  (Numbers)

How many numbers in the interval [1431,9758] are

a.  divisible by 3?

b.  divisible by 5?

c.  divisible by 3 and 5?

d.  divisible by 3 or 5?

2.  (Sets)

Prove that (A ∖ B) ∪ (B ∖ A) = (A ∪ B) ∖ (A ∩ B)

a.  using Venn diagrams,

b.  without Venn diagrams.

[hide answer]

3.  (Alphabets and Words)

Let Σ = {a,b,c} and Ψ = {a,c,e}.

a.  How many words are in the set Σ2 ?

b.  What are the elements of Σ2 ∖ Ψ∗ ?

c.  Is it true that Σ∗ ∖ Ψ∗ = (Σ ∖ Ψ)∗ ?  Why or why not?

(Propositional Logic)

For each of the following formulae, determine all the truth assignments to A, B and C under which the formula is true.

a.  A ∧ (¬C ⇒ (B ∨ ¬A))

b.  (A ∧ ¬C) ⇒ (B ∨ ¬A)

c.  (¬C ⇒ ¬A) ∧ (B ⇒ (A ∧ ¬C))

d.  ¬(C ⇒ A) ∧ (A ∨ (B ∧ ¬C))

4.  (Proving properties of algorithms)

Recall the algorithm for computing the greatest common divisor (gcd) of two positive numbers:

⎧⎪ m

gcd(m,n) = ⎨⎩⎪ gcd(m − n,n)

gcd(m,n − m)
if m = n if m n if m < n
Recall the correctness proof given in class. What needs to be changed to adapt it to the faster version below?

m     if n = 0 gcd(n,mmodn)         if n 0

gcd(m,n) = {

6. Challenge Exercise
A multiplication magic square has the product of the numbers in each row, column and diagonal the same. If the diagram below is filled with positive integers to form a multiplicative magic square, then give the value of y.

 1.  (Numbers)

How many numbers in the interval [1431,9758] are

a.  divisible by 3?

b.  divisible by 5?

c.  divisible by 3 and 5?

d.  divisible by 3 or 5?

2.  (Sets)

Prove that (A ∖ B) ∪ (B ∖ A) = (A ∪ B) ∖ (A ∩ B)

a.  using Venn diagrams,

b.  without Venn diagrams.

[hide answer]

3.  (Alphabets and Words)

Let Σ = {a,b,c} and Ψ = {a,c,e}.

a.  How many words are in the set Σ2 ?

b.  What are the elements of Σ2 ∖ Ψ∗ ?

c.  Is it true that Σ∗ ∖ Ψ∗ = (Σ ∖ Ψ)∗ ?  Why or why not?

(Propositional Logic)

For each of the following formulae, determine all the truth assignments to A, B and C under which the formula is true.

a.  A ∧ (¬C ⇒ (B ∨ ¬A))

b.  (A ∧ ¬C) ⇒ (B ∨ ¬A)

c.  (¬C ⇒ ¬A) ∧ (B ⇒ (A ∧ ¬C))

d.  ¬(C ⇒ A) ∧ (A ∨ (B ∧ ¬C))

4.  (Proving properties of algorithms)

Recall the algorithm for computing the greatest common divisor (gcd) of two positive numbers:

⎧⎪ m

gcd(m,n) = ⎨⎩⎪ gcd(m − n,n)

gcd(m,n − m)
if m = n if m n if m < n
Recall the correctness proof given in class. What needs to be changed to adapt it to the faster version below?

m     if n = 0 gcd(n,mmodn)         if n 0

gcd(m,n) = {

6. Challenge Exercise
A multiplication magic square has the product of the numbers in each row, column and diagonal the same. If the diagram below is filled with positive integers to form a multiplicative magic square, then give the value of y.