COMP9020 Foundations of Computer Science Assignment

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COMP9020 Foundations of Computer Science Assignment - University of New South Wales, Australia

Problem 1 -

(a) List all possible functions f : {a, b, c{ → {0, 1}.

(b) Describe a connection between your answer for (a) and Pow({a, b, c}).

(c) In general, if card(A) = m and card(B) = n, how many:

(i) functions are there from A to B?

(ii) relations are there between A and B?

(iii) symmetric relations are there on A?

Problem 2 -

For x, y ∈ Z we define the set:

Sx,y = {mx + ny : m, n ∈ Z}.

(a) Give five elements of S2,-3.

(b) Give five elements of S12,16.

For the following questions, let d = gcd(x, y) and z be the smallest positive number in Sx,y.

(c) Show that Sx,y ⊆ {n : n ∈ Z and d|n}.

(d) Show that {n : n ∈ Z and z|n} ⊆ Sx,y.

(e) Show that d ≤ z. (Hint: use (c)).

(f) Show that z ≤ d. (Hint: use (d)).

Problem 3 -

We define the operation * on subsets of a universal set U as follows. For any two sets A and B:

A * B := Ac U Bc.

Answer the following questions using the Laws of Set Operations to justify your answer:

(a) What is (A * B) * (A * B)?

(b) Express Ac using only A, * and parentheses (if necessary).

(c) Express ? using only A, * and parentheses (if necessary).

(d) Express A \ B using only A, B, * and parentheses (if necessary).

Problem 4 -

Let Σ = {a, b}. Define R ⊆ Σ* x Σ* as follows:

(w, v) ∈ R if there exists z ∈ Σ* such that v = wz.

(a) Give two words w, v ∈ Σ* such that (w, v) ∉ R and (v, w) ∉ R.

(b) What is R({aba})?

(c) Show that R is a partial order.

Problem 5 -

Show that for all x, y, z ∈ Z:

If x|yz and gcd(x, y) = 1 then x|z.

(Hint: Use the connection between gcd(x, y) and Sx,y shown in Problem 2.).

Attachment:- Foundations of Computer Science Assignment File.rar

Reference no: EM132388524

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