II. Prove that Set Theory is a Model of a Boolean Algebra
The three Boolean operations of Set Theory are the three set operations of union (U), intersection (upside down U), and complement ~. Addition is set union, multiplication is set intersection, and the complement of a set is the set all elements that are in the universal set, but not in the set. The universal set is the set of which all other sets are subsets and the empty set is the set, which has no elements and which therefore all other sets contain. For purposes of this question, let S denote the universal set and Ø the empty set. (Just state the Boolean Algebra equalities of sets below, the proofs are considered self-evident, we do not require Venn diagrams to be written to establish their validity.)
1. State the commutative law of addition: _________________________________________
2. State the associative law of addition: _____________________________________________
3. State the law that says Ø is an additive identity __________________________________
4. State the commutative law of multiplication: ____________________________________
5. State the associative law of multiplication: _______________________________________
6. State the law that says S is a multiplicative identity _____________________________
7. State the distributive law of multiplication: ______________________________________
8. State the distributive law of addition: _____________________________________________
9. State the Boolean Algebra property x + ˜ x = 1 in terms of a set A.
10. State the Boolean Algebra property x • ˜ x = 0 in terms of a set A.
The above ten properties are necessary and sufficient conditions to prove that Set Theory is indeed a model of a Boolean algebra.
11. In Set Theory the difference of two sets, A and B is defined as:
A - B = { s | s belongs to A and s does not belong to B }
Define the difference of two sets A and B, using the basic operations of set theory: union, intersection, and complement.
A - B =
12. In terms of an Abstract Boolean Algebra, for two elements x and y define the difference, x - y using the basic operations +, •, and ~ of Boolean Algebra, using the definition from Set Theory as your guide.
x - y
13. In Boolean Algebra rewrite the expression x - (y + z) using only the basics operations of ~ , • and +.
x - ( y + z ) =
14. Using the results of Boolean Algebra in problem 13 above, rewrite the set theoretic expression of A - ( B U C ) using only the basics operations of set theory : union, intersection, and complement.
A - ( B U C ) =