Answer Happy

Let p(x, y) represent the statement, "Square z is adjacent to square y," and let q(z) represent, "Square z is white." For the purposes of this question: • Squares x, y, z will be among the 3 x 3 grid of squares shown in the figure. • Two squares are adjacent if they share exactly one common side. • Each square has a colour: either white or gray. The square marked with an z has a colour, but it has not been provided to you. You will determine its colour in part (b). (a) [4 marks] Write a plain-English version of the following statements: 3x 3y p(x, y) Aq(x) ^ q(y) 3x Vy -q(r) ^ [p(x,y) →q(y)] (b) [3 marks] If is it known that every gray square is adjacent to at least two white squares, then what is the colour of r? Explain why using exactly one English sentence. (c) [5 marks] Prove or disprove: Vx y q(z) → [p(x, y) ^-q(y)]