The notation and definitions which we are introducing in this chapter will remain valid throughout chapters 2–5 Chapter 6 will use a somewhat different terminology.

Let X = {x,y,z, …} denote the set of all conceivable social states and let N = {1, …, n} denote a finite set individuals or voters (n ≥ 2). Let R stand for a binary relation on X; R is a subset of ordered pairs in the product X × X. We interpret R as a preference relation on X. Without any index, R refers to the social preference relation. When we speak of individual i's preference relation we simply write Ri. The fact that a pair (x, y) is an element of R will be denoted xRy; the negation of this fact will be denoted by ¬xRy. R is reflexive if for all x Є X: xRx. R is complete if for all x,y Є X,x ≠ y: xRy or yRx. R is said to be transitive if for all x,y,z Є X: (xRy ∧ yRz) → xRz. The strict preference relation (the asymmetric part of R) will be denoted by P: xPy ↔ [xRy ∧ ¬yRx]. The indifference relation (the symmetric part of R) will be denoted by I: xIy ↔ [xRy ∧ yRx]. We shall call R a preference ordering (or an ordering or a complete preordering) on X if R reflexive, complete and transitive.