--%>

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 : 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 : Define Big-O notation Big-O notation :

    Big-O notation: If f(n) and g(n) are functions of a natural number n, we write f(n) is O(g(n)) and we say f is big-O of g if there is a constant C (independent of n) such that f

  • Q : Ordinary Differential Equation or ODE

    What is an Ordinary Differential Equation (ODE)?

  • Q : Who firstly use the finite-difference

    Who firstly use the finite-difference method?

  • Q : Problem on mixed-strategy equilibrium

    Assume three Offices (A, B, & C) in downtown,  simultaneously decide whether to situate in a new Building. The payoff matrix is illustrated below. What is (are) the pure stratgy Nash equilibrium (or equilibria) and mixed-strtegy equilibrium of the game?

  • Q : Mathematical and Theoretical Biology

    Mathematical and theoretical biology is an interdisciplinary scientific research field with a range of applications in the fields of biology, biotechnology, and medicine. The field may be referred to as mathematical biology or biomathematics to stress the mathematical

  • Q : Row-echelon matrix Determine into which

    Determine into which of the following 3 kinds (A), (B) and (C) the matrices (a) to (e) beneath can be categorized:       Type (A): The matrix is in both reduced row-echelon form and row-echelon form. Type (B): The matrix

  • Q : Theorem-Group is unique and has unique

    Let (G; o) be a group. Then the identity of the group is unique and each element of the group has a unique inverse.In this proof, we will argue completely formally, including all the parentheses and all the occurrences of the group operation o. As we proce

  • Q : Problem on augmented matrix Consider

    Consider the following system of linear equations.  (a) Write out t

  • Q : Elasticity of Demand For the demand

    For the demand function D(p)=410-0.2p(^2), find the maximum revenue.