Tuesday 10 July 2012

Postulates (continued)


– Inverse. For a set with an identity element with respect to a
binary operation, the set is said to have an inverse if for every
element of the set the following property holds:
» x * y = e
• The additive inverse of element a is –a and it defines
subtraction, since a + (–a) = 0. Multiplicative inverse of
a is 1/a and defines division, since a.1/a = 1
– Distributive Law. * is said to be distributive over . when
» x * ( y · z) = (x * y) · (x * z)
– Note: * + and . are binary operators. Binary operator + defines
addition and binary operator . defines multiplication
• Two-value Boolean algebra is defined by the:
– The set of two elements B={0, 1}
– The operators of AND (·) and OR (+)
– Huntington Postulates are satisfied
Huntington Postulates:

Boolean algebra is an algebraic structure defined by a set
of elements, B , together with two binary operators, + and .
, provided that the following (Huntington) postulates are
satisfied:
1. Closure.
a) with respect to the binary operation OR (+); c=x+y
b)with respect to the binary operation AND (·); c=x.y
2. Identity.
a) with respect to OR (+) is 0:
􀂙x + 0 = 0 + x = x, for x = 1 or x = 0
b)with respect to AND (·) is 1:
􀂙x · 1 = 1 · x = x, for x = 1 or x =0
3. Commutative Law.
a) With respect to OR (+):
􀂙x + y = y + x
b)With respect to AND (·):
􀂙x · y = y · x

4C.Doistnribtuitinveu Leawd.…
a) with respect to the binary operation OR (+):
x + (y · z) = (x + y) · (x + z) + is distributive over .
b)with respect to the binary operation AND (·):
x · (y + z) = (x · y) + (x · z) . is distributive over +
5.Complement. For every element x, that belongs to B, there also
exists an element x’ (complement of x) such that:
a) x + x’ = 1, for x = 1 or x = 0
b)x · x’ = 0 , for x = 1 or x = 0
6. Membership.
There exists at least two elements, x and y, of the set such that
x ≠ y.
0 ≠ 1
Notes on Huntington Postulate 

Comparing Boolean algebra with arithmetic and ordinary
algebra, we note the following differences:
• The associative law is not listed but it can be derived from
the existing postulates for both + and . operations.
• The distributive law of + over . i.e.,
x+(y . z) = (x + y) . (x + z)
is valid for Boolean algebra but not for ordinary algebra.
• Boolean algebra doesn’t have inverses (additive or
multiplicative) therefore there are no operations related to
subtraction or division.
• Postulate 5 defines an operator called complement that is
not available in ordinary algebra.
• Boolean algebra deals with only two elements, 0 and 1



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