Homework 4 -- Sets I

1. Let A = {a, b, c, {d}, {a,c}}. Which of the following are true?

   ____	a is an element of A
   ____	a is a subset of A
   ____	{a} is an element of A
   ____	{a,b} is an element of A
   ____	{a,c} is an element of A
   ____	{a,b} is a subset of A
   ____	{a,c} is a subset of A
   ____	{d} is a subset of A
   ____	{{d}} is a subset of A
   ____	{a,{d}} is a subset of A
   ____	{ } is an element of A
   ____	{ } is a subset of A
   ____	{{ }} is a subset of A
   ____	The cardinality of A is 6.
   

2. Find two finite sets A and B such that A is a subset of B and B is a subset of A.

3. Find the power set of {3, {1,2}, 5}.

4. List five elements of the set {x | x = 3k - 4 and k is a natural number}.

5. Draw 2-set Venn diagrams to illustrate each of the following:
   • B-A
   • (A-B) union (B-A)
   • A intersect (B complement)
6. Draw 3-set Venn diagrams to illustrate each of the following:
   • (C intersect (A union B)) complement
   • (B intersect C)-(A complement)
   • (B - C) union (A intersect B)
7. Given A={1,5,7,13,15} and B={2,3,5,11,15}, find each of the following:
   • A union B
   • A intersect B
   • A-B
   • B-A

   Throughout this homework, ~ is used to mean "not".

8. Symbols:

   ____    is our textbook symbol for  negation
   ____	is our textbook symbol for  conjunction
   ____	is our textbook symbol for  implication
   ____    is our textbook symbol for  disjunction
   

9. Which of the following are true? Check the true ones.

   ____    Maine is north of Florida or Georgia is east of Texas.	
   ____   	Florida is north of Maine or Texas is east of Georgia.	
   ____    Maine is north of Florida and Georgia is east of Texas.	
   ____    Maine is north of Florida and Georgia is west of Texas.	
   ____    Maine is north of Florida or Georgia is west of Texas.	
   ____    Maine is south of Florida and Georgia is west of Texas.	
   ____	~(Maine is south of Florida)
   ____	~(Florida is south of Maine or Texas is east of Georgia)
   

10. Which of the following are true? Check the true ones.

    ____	If 1+1 = 2 then 2+2 = 4.
    ____	If 1+1 = 2 then 2+2 = 8.
    ____	If 1+1 = 7 then 2+2 = 3.
    ____	If 1+1 = 7 then 2+2 = 4.
    

11. Complete the following truth table. Use T for true and F for false.

      A    B    A -> B   ~A   ~B    ~A v ~B   (A -> B) <-> (~A v ~B)
    ------------------------------------------------------------------
        |     |        |     |     |          |               
        |     |        |     |     |          |         
        |     |        |     |     |          |         
        |     |        |     |     |          |         
    

12. Complete the following truth table. Use T for true and F for false.

      A    B      C    A V C   (A V C)^ B   ~A     (A V C) ^ B) -> ~A
    ------------------------------------------------------------------
        |     |      |       |             |     |               
        |     |      |       |             |     |         
        |     |      |       |             |     |         
        |     |      |       |             |     |         
        |     |      |       |             |     |         
        |     |      |       |             |     |         
        |     |      |       |             |     |         
        |     |      |       |             |     |         
    

13. Circle the letter corresponding to the contrapositive of the following statement:

    If Smoky is a cat, then Rex is not a dog. 
    
    a.	If Rex is a dog, then Smoky is not a cat.
    b.	If Rex is not a dog, then Smoky is a cat.
    c.	Rex is a dog, if Smoky is not a cat.
    d.	If Rex is a cat, then Smoky is not a dog.  
    

14. In the last problem, which other two statements of a,b,c,d were equivalent? _______

15. What do the terms tautology and contradiction mean? Give an example (using truth variables) of each.