By Dexter J. Booth (auth.)

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**Example text**

Hence the left-hand proposition can be written as (q A r) v p v (p A q) v (p A q A r) = Again, because p v (p A q) p and p v (p A q A r) the truth function can be written as (q A r) v p =p v (q A =p by the absorption law r) using the commutative law. Hence we have reduced (p v q) A (p v r) to the equivalent and simpler form p v (q A r). This will be recognized as the distributive law that we met previously. Demorganization Every compound proposition has an associated, negated compound proposition that is found by a process of negation known as demorganizing.

Accordingly NOT p is TRUE thereby causing the output: Despatch. (b) Because Balance := 130 and CustomerCredit := 100 then Balance > 125 is TRUE and customercredit = 100 is TRUE. Consequently p is TRUE, being a conjunction of two true propositions. Accordingly NOT p is FALSE thereby causing the output: Do not despatch. (c) Because Balance := 100 and Customercredit := 90 then Balance > 125 is FALSE and customerCredit = 100 is FALSE. Consequently pis FALSE, being a conjunction of two false propositions.

The state of switch B is irrelevant. 10 Reduce the following compound propositions to simpler forms using the laws of logic: (a) The applicant is at least 18 years old and a student or the applicant is at least 18 years old and unemployed. (b) The customer is either given credit or he is a new customer and the customer is either given credit or he has a high-risk credit rating. Solution: (a) Let p =The applicant is at least 18 years old, q =The applicant is a student and r = The applicant is unemployed.