Practice Homework 4 -- Sets I

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

   _x__	a is an element of A
   ____	a is a subset of A
   ____	{a} is an element of A
   _x__	{a} is a subset of A
   _x__	{ } is an element of A
   _x__	{ } is a subset of A
   _x__	{{ }} is a subset of A
   ____	The cardinality of A is 1.
   _x__	The cardinality of A is 2.
   

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

   A = { } and B = {{ }}, or
   A = {1} and B = {1, {1}}
   

3. Find the power set of {a, {a}}.

   P({a,{a}}) = {{ }, {a}, {{a}}, {a,{a}}}
   

4. List five elements of the set {x | x=3k+5 and k is a natural number}.

   5, 8, 11, 14, 17, 20, 23, ...
   

5. Draw 2-set Venn diagrams to illustrate each of the following:
   • A-B
   • (A union B)-(B-A)
   • A intersect B
   
6. Draw 3-set Venn diagrams to illustrate each of the following:
   • (B union C)-(A intersect C)
   • (A intersect B)-C
   • (B - C) union (C - B) union A
   
7. Given A={1,2,4,5,7,9} and B={2,3,4,5,6}, find each of the following:
   • A union B = {1,2,3,4,5,6,7,9}
   • A-B = {1,7,9}
   • B-A = {3,6}
   • A intersect B = {2,4,5}

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

8. Matching:

   _b__	and		a.  negation
   _d__	or		b.  conjunction
   _a__	not		c.  implication
   _c__	if-then		d.  disjunction
   

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

   _x__	3 > 2 and 7 < 8
   ____	3 > 2 and 8 < 7
   ____	2 > 3 and 7 < 8
   _x__	2 > 3 or 7 < 8
   _x__	3 > 2 or 8 < 7
   ____	2 > 3 or 8 < 7
   _x__	~(2 > 3)
   ____	~(3 > 2)
   

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

    ____	If 3 * 2 = 6 then 4 + 3 = 8.
    _x__	If 3 * 2 = 6 then 5 > 4.
    _x__	If 3 * 2 = 7 then 4 + 3 = 8.
    _x__	If 3 * 2 = 7 then 4 + 3 = 7.
    

11. Complete the following truth table. Use T for true and The purpose of the table is to show that an if-then and its contrapositive are logically equivalent, but an if-then is not equivalent to either its converse or its inverse.

      A    B    A -> B   ~B    ~A    ~B -> ~A    B -> A    ~A -> ~B
    ------------------------------------------------------------------
     T  |  T  |   T    |  F  |  F  |     T     |    T    |   T  
     T  |  F  |   F    |  T  |  F  |     F     |    T    |   T 
     F  |  T  |   T    |  F  |  T  |     T     |    F    |   F 
     F  |  F  |   T    |  T  |  T  |     T     |    T    |   T 
    

12. Although an if-then is not logically equivalent to either its converse or its inverse, what can you say about the converse compared to the inverse?

    They are logically equivalent.
    
    

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

    If bells ring then angels sing.
    
    a.	If bells do not ring then angels do not sing.
    b.	If angels sing then bells ring.
    c.<---	If angels do not sing then bells do not ring.
    d.	If bells do not ring then angels sing.