1158107514

# 1158107514 - Boolean Algebra 1. 3. 5. 7. 9. X+0= X X+1 =1...

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CEG 260, Meilin Liu 1 1. 3. 5. 7. 9. 11. 13. 15. 17. Commutative Associative Distributive DeMorgan’s 2. 4. 6. 8. X . 1 X = X . 0 0 = X . X X = 0 = X . X Boolean Algebra 10. 12. 14. 16. X + Y Y + X = ( X + Y ) Z + X + ( Y Z ) + = X ( Y + Z ) XY XZ + = X + Y X . Y = XY YX = ( XY ) Z X ( YZ ) = X + YZ ( X + Y )( X + Z ) = X . Y X + Y = X + 0 X = + X 1 1 = X + X X = 1 = X + X X = X

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CEG 260, Meilin Liu 2 The identities above are organized into pairs. These pairs have names as follows: 1-4 Existence of 0 and 1 5-6 Idempotence 7-8 Existence of complement 9 Involution 10-11 Commutative Laws 12-13 Associative Laws 14-15 Distributive Laws 16-17 DeMorgan’s Laws Some Properties of Identities & the Algebra The dual of an algebraic expression is obtained by interchanging + and · and interchanging 0’s and 1’s. The identities appear in dual pairs. When there is only one identity on a line the identity is self-dual , i. e., the dual expression = the original expression.
CEG 260, Meilin Liu 3 Boolean Operator Precedence The order of evaluation in a Boolean expression is: 1. Parentheses 2. NOT 3. AND 4. OR

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CEG 260, Meilin Liu 4 Basic Identities: Operations with 0 and 1 1. X+0 = X X 0 Y=X X 0 C 0 0 0 1 0 1 3. X+1 = 1 X 1 Y=1 X 1 C 0 1 1 1 1 1 X 0 Y=0 4. X·0 = 0 X 0 C 0 0 0 1 0 0 X 1 Y=X 2. X·1 = X X 1 C 0 1 0 1 1 1 The dual of an algebraic expression is obtained by interchanging OR and AND operations and replacing 1’s by 0’s and 0’s by 1’s. NOTE : the dual of an expression is not equal to the original expression, so a dual can not be used to replace the original expression. If two expressions are equal then their duals are equal.
CEG 260, Meilin Liu 5 Basic Identities: Idempotent Laws 5. X+X = X X X Y=X X X Y X X Y=X 6. X·X = X X X Y

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## This note was uploaded on 05/03/2010 for the course CEG 260 taught by Professor Staff during the Spring '08 term at Wright State.

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1158107514 - Boolean Algebra 1. 3. 5. 7. 9. X+0= X X+1 =1...

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