number_system

# 2 3 4 5 the expression is scanned from left to right

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Unformatted text preview: ad as Does A + B • C mean (A + B) • C or A + (B • C)? The two generate different values for A = 1, B = 0, and C = 0, for then we have (1 + 0) • 0 = 0 and 1 + (0 ■ 0) = 1, which differ. Hence it is necessary to define operator precedence in order to correctly evaluate boolean expressions. The precedence of Boolean operators is as follows: 1. 2. 3. 4. 5. The expression is scanned from left to right. Expressions enclosed within parentheses are evaluated first. All complement (NOT) operations are performed next. All '•' (AND) operations are performed after that. Finally, all'+' (OR) operations are performed in the end. So according to this precedence rule, A + B • C means A + (B • C). Similarly for the expression A • B, the complement of A and B are both evaluated first and the results are then ANDed. Again for the expression [A + BJ, the expression inside the parenthesis (A + B) is evaluated first and the result so obtained is then complemented. Postulates of Boolean Algebra Postulate 1: (a) A = 0 if and only if A is not equal to 1 (b) A = 1 if and only if A is not equal to 0 Postulate 2: (a) x + 0 = x (b) x . 1 = x Postulate 3: Commutative Law (a) x + y = y + x (b) x • y = y • x Postulate 4: Associative Law (a) x + (y + z) = (x + y) + z (b) x . (y • z) = (x • y) • z Postulate 5: Distributive Law (a) x • (y + z) = x • y + x • z (b) x + y . z = (x + y) • (x + z) Postulate 6: (a) x + x = 1 (b) x • x = 0 The postulates listed above are the basic axioms of the algebraic structure and need no proof. They are used to prove the theorems of Boolean algebra. The Principle of Duality In Boolean algebra, there is a precise duality between the operators • (AND) and + (OR), and the digits 0 and 1. For instance, let us consider Figure 6.4. We can see that the second row of the table is obtainable from the first row and vice-versa simply by interchanging '+' with '•' and '0' with ' 1'. This important property is known as the principle of duality in Boolean algebra. Column 1 Column Column 3 2 Row 1 1 + 1 = 1...
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## This document was uploaded on 04/07/2014.

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