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I forgot the name of a law in Boolean algebra, and I can't think of how to ask this question to a search engine. It's the law that states that the disjunction of a variable with the conjunction of its inverse with another variable is equal to the disjunction of the two variables

In algebraic terms:

$x_1 + (\neg(x_1) \cdot x_2) = x_1 + x_2$

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It is sometimes called, Complement Absorption, and is a shorthand for three steps of Distribution, Complementation, and Identity

$\begin{align}a+(\bar a\cdot b) & = (a+\bar a)\cdot(a+b) & \text{Distribution of Disjunction over Conjunction} \\ & = 1\cdot(a+b) & \text{Disjunction of Complements} \\ & = (a+b) & \text{Conjunctive Identity} \end{align}$

Similarly we have:

$\begin{align}a\cdot(\bar a + b) & = (a\cdot\bar a)+(a\cdot b) & \text{Distribution of Conjunction over Disjunction} \\ & = 0+(a\cdot b) & \text{Conjunction of Complements} \\ & = (a\cdot b) & \text{Disjunctive Identity} \end{align}$

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It's just an application of the axioms of distributivity, complements, and identity: $$x_1+(\neg x_1\cdot x_2)=(x_1+\neg x_1)\cdot(x_1+x_2)=1\cdot(x_1+x_2)=x_1+x_2$$which follows from $a+bc=(a+b)(a+c),a\cdot\neg a=1,$ and $1a=a$

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