1. in how many ways can a particle move in the xy-plane from the point (2,3)
to the point (6,9) if the moves that are allowed are the form:
(R):(x,y)→(x+1,y) ; (U):(x,y)→(x,y+1) ; (V):(x,y)→(x+1,y+1)?
(b)how many of the paths in part (a) do not use the path from (3,4) to
(3.5) to (4,5) to (4,6)?
2. Let p(x),q(x), and r(x) denote the following open statements.
p(x): x^2-8x+15= 0; q(x): x is odd; r(x): x > 0
for the universe of all integers, determine the truth or falsity of each of
the following statements.
(a) for all x [q(x) → p(x)] (b) exist x [p(x) → (q(x)^r(x))]
(c) for all x [┐q(x) → ┐p(x)] (d) for all x [p(x)ˇq(x)] → r(x)]
3. (a) How many rows are needed for the truth table of the compound statement
(pˇ┐q) ←→ [(┐rˇs) → t], where p,q,r,s, and t are primitive
statements?
(b) let P1, P2, ..., Pn denote n primitive statements. Let p be a compound
statement that contains at least one of Pi, for 1<=i<=n and p contains
no other primitive statement. How many rows are needed to construct the
truth table for p?
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