Show that p ⋁ (q ⋀ r) and (p ⋁ r) ∧ (p ⋁ r) are logically equivalent. This is the distributive law of disjunction over conjunction.
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Distributive law of disjunction over conjunction has the form
Let we prove this identity
We must conider all cases for p,q,r
1) Let p=1
Left part of the equation has the form
because for any X. be the properties of 1.
Right part equals to
Thus if p=1 both parts of equation eqaul to 1 therefore the identity is true
for all possible values q,r .
Now let be p=0.
In this case left part of identity equals to
because by the properties of 0.
Right part equals to =
Thus in all possible cases p=0 and p=1 left and right parts are equal identically, so identity is proved.
Given equality p ⋁ (q ⋀ r) and (p ⋁ r) ∧ (p ⋁ r) contans mistake because if p=0,r=1 right part of it equals
but left part eqals to
and we must have q=1 but this is not necessary and may be q=0 and so given identity is erroneous.