--%>
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 : State Fermat algorithm The basic Fermat

    The basic Fermat algorithm is as follows: Assume that n is an odd positive integer. Set c = [√n] (`ceiling of √n '). Then we consider in turn the numbers c2 - n; (c+1)2 - n; (c+2)2 - n..... until a perfect square is found. If 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 : Solve each equation by factoring A

    A college student invested part of a $25,000 inheritance at 7% interest and the rest at 6%.  If his annual interest is $1,670 how much did he invest at 6%?  If I told you the answer is $8,000, in your own words, using complete sentences, explain how you

    		•
  • Q : Global And Regional Economic Development

    The Pharmatec Group, a supplier of pharmaceutical equipment, systems and services, has its head office in London and primary production facilities in the US. The company also has a successful subsidiary in South Africa, which was established in 1990. Pharmatec South A

    		•
  • Q : The mean of the sampling distribution

    1. Caterer determines that 87% of people who sampled the food thought it was delicious. A random sample of 144 out of population of 5000 taken. The 144 are asked to sample the food. If P-hat is the proportion saying that the food is delicious, what is the mean of the sampling distribution p-hat?<

    		•
  • Q : Explain trading of call options Explain

    Explain trading of call options.

    		•
  • Q : Pig Game Using the PairOfDice class

    Using the PairOfDice class design and implement a class to play a game called Pig. In this game the user competes against the computer. On each turn the player rolls a pair of dice and adds up his or her points. Whoever reaches 100 points first, wins. If a player rolls a 1, he or she loses all point

    		•
  • Q : Properties of a group How can we say

    How can we say that the pair (G, o) is a group. Explain the properties which proof it.

    		•
  • Q : Breakfast program if the average is

    if the average is 0.27 and we have $500 how much break fastest will we serve by 2 weeks

    		•
  • Q : Theorem-G satis es the right and left

    Let G be a group. (i) G satis es the right and left cancellation laws; that is, if a; b; x ≡ G, then ax = bx and xa = xb each imply that a = b. (ii) If g ≡ G, then (g-1)

    		•