Expert: Josh - 2/1/2009

Question
1. Let U= (q,r,s,t,u,v,w,x,y,z) A= (q,s,u,w,y) B= (q,s,y,z) C= (v,w,x,y,z) List the elements in the set (AUB)'   

2. Find n (A) for the set A= (x/x is the day of the week)   

3. Find the number of subsets of the set x/x is a day of the week)   

4. Determine if the argument is valid or a fallacy. Give a reason to justify answer. If I'm hunry, then I will eat. I'm not hungry I will not eat   

5. Write a conclusion that yeilds a valid argument. All birds have wings. None of my pets are birds. All animals with wings can flap them.   

6. Let p represent the statement "Jim plays football," andlet q represent the statement, "michael plays basketball." Convert the compound statement into symbols.  Neither Jim plays football nor Michael plays basketball.   

7. Rewrite the statement in the form "if p, then q" I will be happy only ifthey call   

8. Write the inverse of the stement: All cats catch birds.   

9. Solve the problem p^q is true, what can you conclude about the truth of p and q?

Answer
1. (A U B) describes the union of set A and B. It includes all elements found in either set A or set B. Answer: (q,s,u,w,y,z).

2. n (A) represents the cardinality (or size) of the set A. First, you need to note how many days are in a week, because this is how the set A is defined. Then, the number of elements in A (in other words, its cardinality) is equal to the number of days in a week.

3. A subset is by definition any collection of elements chosen from the original set A that forms only a portion of A. If we use "1","2","3","4","5","6","7" to denote Monday through to Sunday, and let A={1,2,3,4,5,6,7}, then taking out any number of elements from the set A will produce a subset B with the size n(B) less than n(A).

Consider the number of ways in which such a subset can be made.
There are C(7,6)=7!/(6!1!) ways of selecting six elements from set A (note: the order of selection is irrelevant). Here, 7! is the factorial notation. e.g., n!=n*(n-1)*(n-2)*....*2*1; 6!=6*5*4*3*2*1.
The number of possibilities is the sum of C(7,p) from p=1 to p=6 if we consider only proper subsets. Don't forget the empty set as well.

4. Represent "hunger" => "will eat" with A -> B.
  Its inverse (~A imply ~B) does not necessarily follow.

5. Abstraction
  All birds have wings: "birds" => "have wings"  (A -> B)
  All animals with wings can flap them: "have wings" => "can flap" (B -> C)
None of my pets are birds: "my pets" AND "not birds" (p in ~A)
  Deduce: "birds" => "can flap" (Transitive property: A -> C)
  note: A and ~A are mutually exclusive, they together span the entire space. You may conclude that "my pets cannot flap their wings" (p in ~C)

6. Neither 1st proposition (p), nor 2nd proposition (q) is equivalent to (NOT p) AND (NOT q)

7. If they call, then I will be happy.

8. see http://regentsprep.org/Regents/Math/relcond/Linvers.htm

9. p^q = TRUE if and only if p=TRUE and q=TRUE

Jacqui,

May I draw your attention to my guidelines you might have overlooked when you sent in these questions. You should attempt the problems yourself and think about these questions before asking. I would be happy to help if your problem stems from not understanding the concepts and you show that you have at least attempted the work. I cannot help you if you can't be bothered learning the basics from school or your textbook; fully expecting someone to do it for you.