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# Boolean Algebra

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Boolean Algebra Homework Help Required

Translating formal propositions
¬ = not &#8707; =there exists &#8744; = or &#8801; equivalent
&#8704;= for all &#8658; =implies &#8743; = and &#8660; 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.

¬&#8704;x in X[p(x) &#8658;q(x)]
&#8801;¬&#8704;x in X[¬p(x) &#8744;q(x)] (rewriting &#8658;)
&#8801;&#8707;x in X[¬(¬p(x) &#8744;q(x))] (2)
&#8801;&#8707;x in X[p(x) &#8743;¬q(x)] (3)

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

(ii) ¬&#8707;x in X[p(x) &#8743;¬q(x)] &#8801;&#8704;x in X[¬p(x) &#8744;q(x)]

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

hot &#8658;(cloudy &#8744;¬wet )
hot &#8744;cloudy
¬cloudy

and its aim is to deduce the conclusion ¬wet.

Argument

hot &#8658;(cloudy &#8744;¬wet ) (1), premise
hot &#8744;cloudy (2), premise
¬cloudy (3), premise
hot (4), ?? 3, 2
cloudy &#8744;¬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 &#8744;; Transitivity; Addition, Conjunction; Simplification.

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

https://brainmass.com/math/boolean-algebra/boolean-algebra-174112

#### Solution Preview

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Boolean Algebra Homework Help Required

Translating formal propositions
¬ = not &#8707; =there exists &#8744; = or &#8801; equivalent
&#8704;= for all &#8658; =implies &#8743; = and &#8660; 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.

¬&#8704;x in X[p(x) &#8658;q(x)]
&#8801;¬&#8704;x in X[¬p(x) ...

\$2.19