Discrete Mathematics Answers

If A={1,3,5,7} and B={1,3,7}. Is set B a proper subset of set A? Explain.

Consider the following premises:

1.A-->(B-->A) is a theorem of proportional calculus for all statements A and B.

2.suppose then that the following are the temporary axioms.

a)w

b)y

c)y-->z

Using the logical rules of inference, modus ponens and hypothetical syllogism, show that x-->z is diducible from the given premises.

Let O be the set of odd numbers and O’ = {1, 5, 9, 13, 17, ...} be its subset. Define

the bijections, f and g as:

f : O "\\to" O’, f(d) = 2d - 1, "\\forall" d "\\in" O.

g : "\\Nu" "\\to" O, g(n) = 2n + 1, "\\forall" n "\\isin" "\\Nu" .

Using only the concept of function composition, can there be a bijective map from "\\Nu"

to O’? If so, compute it. If not, explain in details why not.

Consider the statement form (P↓Q)↓R.

Now, find a restricted statement form logically equivalent to it, in

a) Disjunctive normal form (DNF).

b) Conjunctive normal form (CNF).

Consider the statement form

(P \downarrow Q) \downarrow  R

Now, find a restricted statement form logically equivalent to it, in

a) Disjunctive normal form (DNF).

b) Conjunctive normal form (CNF).

 Show that ∃xP(x) ∧ ∃xQ(x) and ∃x(P(x) ∧ Q(x)) are not logically equivalent.

 Show that ∃x(P(x) ∨ Q(x)) and ∃xP(x) ∨ ∃xQ(x) are logically equivalent.

 Determine whether ∀x(P(x) ↔ Q(x)) and ∀x P(x) ↔ ∀xQ(x) are logically equivalent. Justify your answer.

 Determine whether ∀x(P(x) → Q(x)) and ∀xP(x) → ∀xQ(x) are logically equivalent. Justify your answer.