**Unit 5 Exercises**

**1.** Use truth table to verify the following equivalences.

a) *p*
**F**
*F*

* *b) *p *
**T**
**T**

** **c) *p*
*p*
*p*

**2.** Use truth tables to verify the distributive law *p*
(*q*
*r*)
(*p*
*q*)
(*p*
*r*).

**3.** Show that each of the following implications is a tautology
without using truth tables.

a) *p*
(*p*
*q)*

* *b) (*p*
*q*)
(*p*
*q*)

c)
(*p* *q*)
*q*

**4.** Verify the following equivalences, which are known as the absorption
laws.

a) [* p*
(*p*
*q*) ]
*p*

b) [* p*
(*p*
*q*) ]
*p*

**5.** Find the dual of each of the following propostions.

*p**q**r*- (
*p**q**r*)*s* - (
*p***F**) (*q***T**)

**6.** Find a compound proposition involving the propositions
*p*,*q*, and *r* that is true when exactly one of *p*,
*q*, and *r*is true and is false otherwise. (*Hint*:
Form a disjunction of conjunctions. Include a conjunction for each combination
of values for which the propostion istrue. Each conjunction should include each
of the three propositions or theirnegations.)