--%>
Assignment Help - homework help

Set Theory & Model of a Boolean Algebra

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 ) = 

   Related Questions in Mathematics

  • Q : First-order formulas over the

    Consider the unary relational symbols P and L, and the binary relational symbol On, where P(a) and I(a) encode that a is apoint and a (sraight) line in the 2-dimensional space, respectively, while On(a,b) encodes  that a is a point, b is a line, and o lies on b.

    		•
  • Q : Where would we be without stochastic

    Where would we be without stochastic or Ito^ calculus?

    		•
  • Q : Containee problem For queries Q 1 and Q

    For queries Q1 and Q2, we say Q1 is containedin Q2, denoted Q1 C Q2, iff Q1(D) C Q2

    		•
  • Q : Relationships Between Data Introduction

    Relationships Between Data - Introduction to Linear Regression Simple Regression Notes If you need guidance in terms of using Excel to run regressions, check pages 1 - 10 of the Excel - Linear Regression Tutorial posted to th

    		•
  • Q : Abstract Algebra let a, b, c, d be

    let a, b, c, d be integers. Prove the following statements: (a) if a|b and b|c. (b) if a|b and ac|bd. (c) if d|a and d|b then d|(xa+yb) for any x, y EZ

    		•
  • Q : Calculus I need it within 4 hours. Due

    I need it within 4 hours. Due time March 15, 2014. 3PM Pacific Time. (Los Angeles, CA)

    		•
  • Q : Problem on inverse demand curves In

    In differentiated-goods duopoly business, with inverse demand curves: P1 = 10 – 5Q1 – 2Q2P2 = 10 – 5Q2 – 2Q1 and per unit costs for each and every firm equal to 1.<

    		•
  • Q : Formulating linear program of a

    A software company has a new product specifically designed for the lumber industry. The VP of marketing has been given a budget of $1,35,00to market the product over the quarter. She has decided that $35,000 of the budget will be spent promoting the product at the nat

    		•
  • Q : Problem on Linear equations Anny, Betti

    Anny, Betti and Karol went to their local produce store to bpought some fruit. Anny bought 1 pound of apples and 2 pounds of bananas and paid $2.11.  Betti bought 2 pounds of apples and 1 pound of grapes and paid $4.06.  Karol bought 1 pound of bananas and 2

    		•
  • Q : Who firstly use the finite-difference

    Who firstly use the finite-difference method?

    		•