MATH221 – Mathematics for Computer Science –  Assignment Two Solved

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Question 1. [4 Marks]

• Find a natural number n such that 2 · 1066 + 1492 ≡ n (mod 1776). Is n unique?
• Show that there is no n ^∈ N such that n ≡ 3 (mod 4) and n ≡ 5 (mod 8).

Question 2. [4 Marks] Identify the shaded regions, using only the union, intersection and complement symbols.

Question 3. [4 Marks] What is the coefficient of x⁵ in the expression (−4 + 3x)¹²? (Do NOT multiply −4 + 3x by itself 12 times.)

Question 4. [8 Marks] Prove or disprove that the following are equivalence relations. If you find one (or both) that is an equivalence relation, write out [−1] and .

• For a,b,c,d ^∈ Z with
• For x,y ∈ R: R = {(x,y) : |x − y| ≤ 1}.