--%>
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 : Use MS Excel to do the computations

    Select a dataset of your interest (preferably related to your company/job), containing one variable and atleast 100 data points. [Example: Annual profit figures of 100 companies for the last financial year]. Once you select the data, you should compute 4-5 summary sta

    		•
  • 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 : Budgeted cash disbursements The ABC

    The ABC Company, a merchandising firm, has budgeted its action for December according to the following information: • Sales at $560,000, all for cash. • The invoice cost for goods purc

    		•
  • Q : Who firstly use the finite-difference

    Who firstly use the finite-difference method?

    		•
  • Q : Uniform scaling what is uniform scaling

    what is uniform scaling in computer graphic

    		•
  • Q : Nonlinear integer programming problem

    Explain Nonlinear integer programming problem with an example ?

    		•
  • Q : Competitive equilibrium 8. Halloween is

    8. Halloween is an old American tradition. Kids go out dressed in costume and neighbors give them candy when they come to the door. Spike and Cinderella are brother and sister. After a long night collecting candy, they sit down as examine what they have. Spike fi

    		•
  • Q : State Prime number theorem Prime number

    Prime number theorem: A big deal is known about the distribution of prime numbers and of the prime factors of a typical number. Most of the mathematics, although, is deep: while the results are often not too hard to state, the proofs are often diffic

    		•
  • Q : Explain Factorisation by trial division

    Factorisation by trial division: The essential idea of factorisation by trial division is straightforward. Let n be a positive integer. We know that n is either prime or has a prime divisor less than or equal to √n. Therefore, if we divide n in

    		•
  • Q : Problem on reduced row-echelon The

    The augmented matrix from a system of linear equations has the following reduced row-echelon form. 280_row echelon method.jpg

    		•