AXIOM 2: (Reflexivity)VxEX, x2x AXIOM 3: (Transitivity)(x y)&(y z) The first assumption says that any two bundles can be compared, the second is trivial, and the third is necessary for any discussions of preference maximization: for if preferences were not transitive, there might be sets of bundles which had no best elements It is useful to extend our notation: We write x >y and say that x is strictly preferred to y. We sometime also write not y 2 x, meaning y is not preferred to x, which is the same as x >y, given completeness We writex-yif(x& y)&(x2 y) and say that x is indifferent to y Examples (a)Finite Set If X is a finite set, then a preference relation on X will partition X into a finite number of subsets such that elements within a subset are all indifferent There will be a strict preference for elements from different subsets (b) Summation Ordering Let X=r Define x a y to mean that∑x≥∑y It is easy to show that this summation ordering is complete, reflective and transitive (c) Lexicographic Ordering Let x=r+ x y if and only if either, there exists some such that xi=yi for i<j and x;>yi; orx= yi for 1≤i≤m Essentially, the lexicographic ordering compares the components one at a time beginning with the first, and determines the ordering based one the first a difference is found This implies that the vector with greatest component is raked the highest The above three axioms are the basic properties of a preference relation. Any relation satisfyin these 3 axioms is called an ordering In order to have a functional representation, we may need a few more axioms(assumptions).(If X is countable, no additional axiom is needed. AXIOM 4:( Continuity) For all y in X, the sets (x: Ia y) and (x: y2 x) are closed sets.It follows that the sets ( x: x>y) and x: y>x) are open sets This assumption is necessary to rule out certain discontinuous behavior It says that, if( x')is a sequence of consumption bundles that are all at least as good as y and if this sequence converges to some bundle x', then x is at least as good as y The key consequence of continuity is as follows: if y is strictly preferred to z and if x is bundle that is close enough to y, then x must be strictly preferred to z E2 AXIOM 2: (Reflexivity)  x  X, x x. AXIOM 3: (Transitivity) (x y) & (y z)  x z. (Note: & = "and") Note: • The first assumption says that any two bundles can be compared, the second is trivial, and the third is necessary for any discussions of preference maximization: for if preferences were not transitive, there might be sets of bundles which had no best elements. It is useful to extend our notation: • We write x  y and say that x is strictly preferred to y. We sometime also write not y x, meaning y is not preferred to x, which is the same as x  y, given completeness. • We write x ~ y if (x y) & (x y) and say that x is indifferent to y. Examples: (a) Finite Set • If X is a finite set, then a preference relation on X will partition X into a finite number of subsets such that • elements within a subset are all indifferent; • There will be a strict preference for elements from different subsets. (b) Summation Ordering: • Let X = R m . • Define x y to mean that . =1 =1  m i i m i i x y • It is easy to show that this summation ordering is complete, reflective and transitive. (c) Lexicographic Ordering • Let X = R m +. • x y if and only if • either, there exists some j such that xi = yi for i < j and xj > yj; • or, xi = yi for 1 i  m. • Essentially, the lexicographic ordering compares the components one at a time beginning with the first, and determines the ordering based one the first a difference is found. • This implies that the vector with greatest component is raked the highest. The above three axioms are the basic properties of a preference relation. Any relation satisfying these 3 axioms is called an ordering. In order to have a functional representation, we may need a few more axioms (assumptions). (If X is countable, no additional axiom is needed.) AXIOM 4: (Continuity) For all y in X, the sets {x: x y} and {x: y x} are closed sets. It follows that the sets {x: x  y} and {x: y  x} are open sets. • This assumption is necessary to rule out certain discontinuous behavior. • It says that, if ( x i ) is a sequence of consumption bundles that are all at least as good as y and if this sequence converges to some bundle x * , then x * is at least as good as y. • The key consequence of continuity is as follows: if y is strictly preferred to z and if x is bundle that is close enough to y, then x must be strictly preferred to z. Examples