# Boolean Algebra

Boolean Algebra Homework Help Required

Translating formal propositions

¬ = not ∃ =there exists ∨ = or ≡ equivalent

∀= for all ⇒ =implies ∧ = and ⇔ if and only if

(a)An argument to show that two propositions are equivalent using the rules of Boolean algebra is given below. Give a rule or rules of Boolean algebra which can be used to justify each of steps (2) and (3) in this argument.

¬∀x in X[p(x) ⇒q(x)]

≡¬∀x in X[¬p(x) ∨q(x)] (rewriting ⇒)

≡∃x in X[¬(¬p(x) ∨q(x))] (2)

≡∃x in X[p(x) ∧¬q(x)] (3)

(b)Using the rules of Boolean algebra show the following. State which rule is used at each step.

(ii) ¬∃x in X[p(x) ∧¬q(x)] ≡∀x in X[¬p(x) ∨q(x)]

(c) Part of a formal argument is given below. This argument has premises

hot ⇒(cloudy ∨¬wet )

hot ∨cloudy

¬cloudy

and its aim is to deduce the conclusion ¬wet.

Argument

hot ⇒(cloudy ∨¬wet ) (1), premise

hot ∨cloudy (2), premise

¬cloudy (3), premise

hot (4), ?? 3, 2

cloudy ∨¬wet (5), ?? 4, 1

(i) For each of steps (4) and (5) in the argument above, say which rule of inference has been used.

(ii) Using the rules of inference, complete a formal argument to reach the conclusion ¬wet .

For easy reference purposes the rules of inference are: Implication; Contraposition; Equivalence; Disjunction; Combining ∨; Transitivity; Addition, Conjunction; Simplification.

For (a) & (b) above the steps presumably relates to De Morgan rules of associativity, negation and absorption.

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Boolean Algebra Homework Help Required

Translating formal propositions

¬ = not ∃ =there exists ∨ = or ≡ equivalent

∀= for all ⇒ =implies ∧ = and ⇔ if and only if

(a)An argument to show that two propositions are equivalent using the rules of Boolean algebra is given below. Give a rule or rules of Boolean algebra which can be used to justify each of steps (2) and (3) in this argument.

¬∀x in X[p(x) ⇒q(x)]

≡¬∀x in X[¬p(x) ...

Free Boolean Algebra

Please help answer the following question.

Using your knowledge of free objects in a category, give a definition of a free Boolean algebra B on a set D. How these compare to free Boolean rings?

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