Introduction
Category : JEE Main & Advanced
Boolean algebra is a tool for studying and applying mathematical logic which was originated by the English mathematician George Boolean. In 1854 he wrote a book “An investigation of the law of thought”, be developed a theory of logic using symbols instead of words. This more algebraic treatment of subject is now called boolean algebra
Definition : A non empty set B together with two operations denoted by \['\vee '\] and \[\,'\wedge '\] is said to be a boolean algebra if the following axioms hold :
(i) For all \[x,y\in B\]
(a) \[x\vee y\in B\] (Closure property for \[\vee \])
(b) \[x\wedge y\in B\] (Closure property for \[\wedge \])
(ii) For all \[x,y\in B\]
(a) \[x\vee y=y\vee x\] (Commutative law for \[\vee \])
(b) \[x\wedge y=y\wedge x\] (Commutative law for \[\wedge \])
(iii) For all x, y and z in B,
(a) \[(x\vee y)\vee z=x\vee (y\vee z)\] (Associative law of\[\vee \])
(b) \[(x\wedge y)\wedge z=x\wedge (y\wedge z)\] (Associative law of \[\wedge \])
(iv) For all x, y and z in B,
(a) \[x\vee (y\wedge z)=(x\vee y)\wedge (x\vee z)\] (Distributive law of \[\vee \] over\[\wedge \])
(b) \[x\wedge (y\vee z)=(x\wedge y)\vee (x\wedge z)\] (Distributive law of \[\wedge \] over \[\vee \])
(v) There exist elements denoted by 0 and 1 in B such that for all \[x\in B\],
(a) \[x\vee 0=x\] (0 is identity for \[\vee \])
(b) \[x\wedge 1=x\] (1 is identity for \[\wedge \])
(vi) For each \[x\in B\], there exists an element denoted by x¢, called the complement or negation of x in B such that
(a) \[x\vee x'=1\]
(b) \[x\wedge x'=0\] (Complement laws)
Principle of duality
The dual of any statement in a boolean aglebra B is the statement obtained by interchanging the operation Ú and Ù, and simultaneously interchanging the elements 0 and 1 in the original statement.
In a boolean algebra, the zero element 0 and the unit element 1 are unique.
Let B be a boolean algebra. Then, for any x and y in B, we have
(a) \[x\vee x=x\] (a¢) \[x\wedge x=x\]
(b) \[x\vee 1=1\] (b¢) \[x\wedge 0=0\]
(c) \[x\vee (x\wedge y)=x\] (c¢) \[x\wedge (x\vee y)=x\]
(d) \[{0}'=1\] (d¢) \[{1}'=0\]
(e) \[({x}'{)}'=x\]
(f) \[(x\vee y{)}'={x}'\wedge {y}'\] (f¢) \[(x\wedge y{)}'={x}'\vee {y}'\]
Important points :
For \[P(A)\], the set of all subsets of a set A, the operations \[\cup \]and \[\cap \] play the roles of \['\vee '\]and \['\wedge '\], A and \[\varphi \] play the role of 1 and 0, and complementation plays the role of ‘¢’.
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