Section 1.3: Propositional Equivalence4. a. The truth table is,pFFFFTTTTqFFTTFFTT(q) r(q)rFTFTFTFTFFTTTTTTFTTTTTTT(qr)FTTTFTTTp(qr)FTTTTTTTAs, the truth values of (q) r and p(qr) agree, hence these are logically equivalent and,(q) r = p(qr) (Proved).b. The truth table is,pFFFFTTTTqFFTTFFTT(q) r(q)rFTFTFTFTFFFFFFTTFFFFFFFT(qr)FFFTFFFTp (qr)FFFFFFFTAs, the truth values of () and () agree, hence these are logically equivalent and,() = () (Proved).8. Using De Morgans laws,a. Let the Boolean variable p and q are,p = Kwame will take a job in industry.q = Kwame will go to graduate school1Hence, the given claim Kwame will take a job in industry or go to graduate school is, (). Thenegation will be, (). Using De Morgans law, it can be written as, (). Hence, the statementwill be, Kwame will not take a job in industry and Kwame will not go to graduate school.b. Let the Boolean variable p and q are,p = Yoshiko knows Java.q = Yoshiko knows calculus.Hence, the claim Yoshiko knows Java and calculus. is, (). The negation is, (). Using DeMorgans laws it can be written as, (). Hence, the statement will be, Yos ...

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