## Try It Now

Work these exercises to see how well you understand this material.

### Exercises

- Prove with truth tables:
*p*∨*q**,**¬**q*⇒*p**p**→**q**,**¬**q*⇒*¬**p*

- Give direct and indirect proofs of:

*a**→**b,**c**→**b,**d**→*(*a*∨*c*)*, d*⇒*b*.- (
*p**→**q*) ∧ (*r**→**s*)*,*(*q**→**t*) ∧ (*s**→**u*)*,**¬*(*t*∧*u*)*, p**→**r*⇒*¬**p*. *p**→*(*q**→**r*)*,**¬**s*∨*p, q*⇒*s**→**r*.*p**→**q**,**q**→**r**,**¬*(*p*∧*r*)*, p**∨**r*⇒*r*.*¬**q**,**p**→**q**,**p*∨*t*⇒*t*

- Are the following arguments valid? If they are valid, construct formal proofs; if they aren’t valid, explain why not.

- If wages increase, then there will be inflation. The cost of living will not increase if there is no inflation. Wages will increase. Therefore, the cost of living will increase.
- If the races are fixed or the casinos are crooked, then the tourist trade will decline. If the tourist trade decreases, then the police will be happy. The police force is never happy. Therefore, the races are not fixed.

- Describe how
*p*_{1}*, p*_{1}*→**p*_{2}*, . . . , p*_{99}*→**p*_{100}⇒*p**100*could be proved in 199 steps.

Source: Al Doerr and Ken Levasseur, http://faculty.uml.edu/klevasseur/ads-latex/ads.pdf

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