Unit 5 Exercises

1. Use truth table to verify the following equivalences.

    a) p and.gif (67 bytes) F eqvT.gif (73 bytes) F

   b) p or.gif (64 bytes) T eqvT.gif (73 bytes) T

   c) p and.gif (67 bytes) p eqvT.gif (73 bytes) p

2. Use truth tables to verify the distributive law p and.gif (67 bytes) (q or.gif (64 bytes) r) eqvT.gif (73 bytes) (p and.gif (67 bytes) q) or.gif (64 bytes) (p and.gif (67 bytes) r).

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

    a) p imp.gif (64 bytes) (p or.gif (64 bytes) q)

    b) (p and.gif (67 bytes) q) imp.gif (64 bytes) (p imp.gif (64 bytes) q)

    c) not.gif (54 bytes) (p imp.gif (64 bytes) q) imp.gif (64 bytes) not.gif (54 bytes) q

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

    a) [ p or.gif (64 bytes) (p and.gif (67 bytes) q) ] eqvT.gif (73 bytes) p

    b) [ p and.gif (64 bytes) (p or.gif (67 bytes) q) ] eqvT.gif (73 bytes) p

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

  1. p or.gif (67 bytes) not.gif (54 bytes) q or.gif (67 bytes) not.gif (54 bytes) r
  2. (p or.gif (67 bytes) q or.gif (67 bytes) r) and.gif (64 bytes) s
  3. (p and.gif 
(64 bytes) F) or.gif (67 bytes) (q and.gif (64 bytes) T)

6. Find a compound proposition involving the propositions p,q, and r that is true when exactly one of p, q, and ris 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.)


Answers for these exercises