values 1 (true) and 0 (false) for variable x. The shading indicates the value of the operation for each combination of regions, with dark denoting 1 and light 0 (some authors use the opposite convention). The three Venn diagrams in the figure below represent respectively conjunction x∧y, disjunction x∨y, and complement ¬x. For conjunction, the region inside both circles is shaded to indicate that x∧y is 1 when both variables are 1. The other regions are left unshaded to indicate that x∧y is 0 for the other three combinations. The second diagram represents disjunction x∨y by shading those regions that lie inside either or both circles. The third diagram represents complement ¬x by shading the region not inside the circle. While we have not shown the Venn diagrams for the constants 0 and 1, they are trivial, being respectively a white box and a dark box, neither one containing a circle. However we could put a circle for x in those boxes, in which case each would denote a function of one argument, x, which returns the same value independently of x, called a constant function. As far as their outputs are concerned, constants and constant functions are indistinguishable; the difference is that a constant takes no arguments, called a zeroary or nullary operation, while a constant function takes one argument, which it ignores, and is a unary operation. Venn diagrams are helpful in visualizing laws. The commutativity laws for ∧ and ∨ can be seen from the symmetry of the diagrams: a binary operation that was not commutative would not have a symmetric diagram because interchanging x and y would have the effect of reflecting the diagram horizontally and any failure of commutativity would then appear as a failure of symmetry. Idempotence of ∧ and ∨ can be visualized by sliding the two circles together and noting that the shaded area then becomes the whole circle, for both ∧ and ∨. To see the first absorption law, x∧(x∨y) = x, start with the diagram in the middle for x∨y and note that the portion of the shaded area in common with the x circle is the whole of the x circle. For the second absorption law, x∨(x∧y) = x, start with the left diagram for x∧y and note that shading the whole of the x circle results in just the x circle being shaded, since the previous shading was