Math 3312 – Homework Assignment 1

Due February 3, 2004

 

Directions: Justify your solution to each problem as appropriate; neatness and organization of your work is important.

 

1.      Consider Theorem 1.3, page 6.

Theorem 1.3:  For any sets A and B, the following are statements are equivalent:

            (i)         A Í B

            (ii)        A Ç B = A

(iii)                A È B = B

Study the solution to problem 1.13 in which a proof of Theorem 1.3 is given of the form “(i) if and only if (ii), and (i) if and only if (iii).”

You are to develop a new proof of Theorem 1.3 of the form “If (i) then (ii); if (ii) then (iii); and if (iii) then (i).”

 

2.  Prove Distributive law 4a

      A  È (B Ç C) = (A È B)  Ç  (A È C)

      (Hint: Study the solution to problem 1.16.)

 

 

3.  Prove the following identity:  (A Ç B ) È ( A Ç B^C ) = A.

      (Hint: Study the solution to problem 1.19.)

 

4.  Consider the class B where

                        B = [ f, {2}, {1, 4}, {0, 2, 5} ]

 

For each of the following statements, determine whether it is true or false; justify your answer for each statement in a complete sentence. (Hint: See the solution to problem 1.3.)

a.       2ÎB

b.       {1, 4}ÎB

c.       {1}ÍB

d.       { {0, 2, 5} }ÍB

e.       fÍB

f.         { f }ÎB