Consider the following statement: There do not exist 2 × 2 matrices E, F with rank(E)=rank(F)= 2 such that E · F = Select whether it is true or false, and give either a proof or counterexample as appropriate. This statement is true Proof: Correct answer, well done. 0 (88). 0

Prove ¬¬A := ¬(¬A) = A ¬(A ^ B) = (¬A) V (¬B) ¬(A V B) = (¬A) ^ (¬B) (Vx € X: E(x)) = (x € X : ¬E(x))

Let f : (0, 1/3 ] → R such that f (x) = x^2.Part 1: Show that f (x) < x.Part 2: Show that f has no fixed point on (0, 1/3 ]. Hint: Assume there were a point c in (0, 1/3 ]such that f (c) = c and derive a contradiction.Part 3: Show that the function f (x) = 1/(1+x^2) from [0, ∞) to [0, ∞) has a fixed point c. Hint: Setf (x) = x and show the resulting equation has a solution in [0, ∞) using the the IVP