# Boolean Algebra **The branch of algebra dealing with binary variables and describing logical operations.** ## Logical operations The **logical operations** are the Boolean analogue of [[Set Algebra#Set operations|set operations]] and are also used in formal logic. In electronics, they are idealised as *logic gates*. - [[Disjunction]] - $X\vee Y$ - $1110$ - [[Conjunction]] - $X\wedge Y$ - $1000$ - [[Negation]] - $\neg X$ - $01$ - [[Exclusive Disjunction]] - $X\oplus Y$ - $0110$ - [[Material Biconditional]] - $X\leftrightarrow Y$ - $1001$ - [[Material Conditional]] - $X\rightarrow Y$ - $1011$ ## Laws The following pairs of laws are all related by the [[Duality#Boolean Algebra|duality]] principle. > [!Identity] > $X\wedge1=X$ > $X\vee 0=X$ > [!Example] Annihilation/Domination > $X\wedge 0=0$ > $X\vee 1=1$ > [!Complementation] > $X\wedge\neg X=0$ > $X\vee\neg X=1$ > [!Example] [[Involution]] > $\neg(\neg X)=X$ > [!Idempotence] > $X\wedge X=X$ > $X\vee X=X$ > [!Commutativity] > $X\wedge Y = Y\wedge X$ > $X \vee Y = Y\vee X$ > [!Associativity] > $X\wedge(Y\wedge Z)=(X\wedge Y)\wedge Z$ > $X\vee(Y\vee Z)=(X\vee Y)\vee Z$ > [!Distributivity] > $X\wedge(Y\vee Z)=(X\wedge Y)\vee(X\wedge Z)$ > $X\vee(Y\wedge Z)=(X\vee Y)\wedge(X\vee Z)$ > [!Absorption] > $X\wedge(X\vee Y) = X$ > $X\vee(X\wedge Y) = X$ > [!Example] Minimisation/Redundancy > $(X\vee Y)\wedge(X\vee\neg{Y})=X$ > $(X\wedge Y)\vee(X\wedge\neg{Y})=X$ > [!Consensus] > $(X\vee Y)\wedge(\neg{X}\vee Z)\wedge(Y\vee Z) = (X\vee Y)\wedge(\neg{X}\vee Z)$ > $(X\wedge Y)\vee(\neg{X}\wedge Z)\vee(Y\wedge Z)=(X\wedge Y)\vee(\neg{X}\wedge Z)$ > [!Example] [[De Morgan's Laws]] > $\neg({X\vee Y})=\neg{X}\wedge\neg{Y}$ > $\neg({X\wedge Y})=\neg{X}\vee\neg{Y}$