Propositional Logic
Using operator properties and other logical equivalences (not
truth tables), prove these statements.
1. ((p→r)∧(q→r)∧(p∨q))→r (tautology)
2. ¬(q→p)∧(p∧q∧s→r)∧p (contradiction)
3. (p→q)∧(p→r)≡p→(q∧r)
1. Prove or disprove: if f : R → R is injective and g : R → R is
surjective then f ◦ g : R → R is bijective.
2. Suppose n and k are two positive integers. Pick a uniformly
random lattice path from (0, 0) to (n, k). What is the probability
that the first step is ‘up’?
Using logical equivalence laws, show that (((p v ~ q) ⊕ p) v ~p)
⊕ (p v ~q) is equivalent to p v q. v = or, ~ = not, ⊕ = exclusive
or (XOR). Please show the steps with the name of the law beside
each step, thanks so much!
For a perceptron algorithm with a fixed learning rate(r), prove
or disprove that if the classes are not linearly separable, it may
not converge. Can you modify the learning rate(r) so that the
iterations converge in expectation? Explain.