Show that p ⋁ (q ⋀ r) and (p ⋁ r) ∧ (p ⋁ r) are logically equivalent. This is the distributive law of disjunction over conjunction.

Distributive law of disjunction over conjunction has the form

$p\lor(q\land r)=(p\lor q)\land(p\lor r)$

Let we prove this identity

We must conider all cases for p,q,r$\in \lBrace 0,1\rBrace$

1) Let p=1

Left part of the equation has the form

$1\lor(p\land r)=1$ because $1\lor X=1$ for any X. be the properties of 1.

Right part equals to$(1∨ q)\land (1\lor r)=1\land 1=1$

Thus if p=1 both parts of equation eqaul to 1 therefore the identity is true

for all possible values q,r$\in \lBrace 0,1 \rBrace$ .

Now let be p=0.

In this case left part of identity equals to $0\lor(q\land r)=q\land r$

because $0\lor X=X$ by the properties of 0.

Right part equals to $(0\lor q)\land(0\lor r)$ =$q\land r$

and $q\land r \equiv q\land r$

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 $(0\lor 1)\land (0\lor 1)=1\land 1=1$

$(0\lor 1)\land (0\lor 1)=1\land 1=1$ but left part eqals to

$0\lor(q\land 1)=0\lor q=q$

and we must have q=1 but this is not necessary and may be q=0 and so given identity is erroneous.