What is the $\lor$ symbol?

Question

In researching the consensus algorithm, I came upon the consensus theorem:

How does the $\lor$ symbol function?

Answer 1

The symbol $\vee$ usually denotes the logical disjunction (the OR truth-functional operator), just as $\wedge$ usually denotes logical conjunction (the AND truth-functional operator).

The empty operator probably refers to logical conjunctions as well, and the upper bar should refer to logical negation.

As suggested by @Joffysloffy, you can map these to operators on variables with values in $\{ 0, 1 \}$, $F$ mapping to $0$ and $T$ mapping to $1$, with (lacking notation rigor) :

$$ x \wedge y = xy $$ $$ x \vee y = x + y - xy $$

(The latter is to be compared with $\left\vert{A \cup B}\right\vert = \left\vert{A}\right\vert + \left\vert{B}\right\vert - \left\vert{A \cap B}\right\vert$)

Answer 2

A Venn diagram might make it clearer how to imagine $\lor$ as the logical or:

The diagram can be used to visualize boolean values as sets. An element may either be member of a set or not.

Three boolean variables lead to eight different areas in theVenn diagram or eight corresponding cases as shown in the following truth table:

In the Venn diagram $xy$ is depicted by the overlap area of circle $x$ and circle $y$. Both have to be true to get a true value for $xy$. That corresponds to [6] and [7] in the table. For $\bar{x}z$ stands the area not in $x$ but in $z$. $yz$ is shown as the area overlapped by $y$ and by $z$.

You can see from the diagram that the Consensus Theorem holds: Right-hand-side and left-hand-side are in fact equivalent.

The $yz$ (= terms [3] + [7]) is covered by $xy \lor \bar{x}z$ (= terms [3]+[6]+[7]) und thus does not add anything to the left-hand-side.

Answer 3

If you interpret $x,y,z$ to take values $0$ (false) and $1$ (true), then you can see the logical AND operator (here logical AND of $x$ and $y$ is written as $xy$) as regular multiplication: $xy=1$ if and only if $x=y=1$. Then the logical OR ($\vee$) can be seen as the maximum function. So $x\vee y=\max\{x,y\}$. Hence $x\vee y=1$ if and only if $x=1$ or $y=1$ (or both).