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.

 Express each of these statements using quantifiers. Then form the negation of the statement, so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase “It is not the case that.”) a) Some old dogs can learn new tricks. b) No rabbit knows calculus. c) Every bird can fly. d) There is no dog that can talk. e) There is no one in this class who knows French and Russian.