a good smoothly as price rose relatively for that good. in Fig. 2 would have either no effect on quantity demanded of either good (at one end of the budget constraint) or would change quantity demanded from one end of the budget constraint to the other. in Fig. 3 would have no effect on equilibrium quantities demanded, since the budget line would rotate around the corner of the indifference curve. Preference relations and utility Choice theory formally represents consumers by a preference relation, and use this representation to derive indifference curves showing combinations of equal preference to the consumer. Preference relations Let be a set of mutually exclusive alternatives among which a consumer can choose. and be generic elements of . In the language of the example above, the set is made of combinations of apples and bananas. The symbol is one such combination, such as 1 apple and 4 bananas and is another combination such as 2 apples and 2 bananas. A preference relation, denoted , is a binary relation define on the set . The statement is described as ' is weakly preferred to .' That is, is at least as good as (in preference satisfaction). The statement is described as ' is weakly preferred to , and is weakly preferred to .' That is, one is indifferent to the choice of or , meaning not that they are unwanted but that they are equally good in satisfying preferences. The statement is described as ' is weakly preferred to , but is not weakly preferred to .' One says that ' is strictly preferred to .' The preference relation is complete if all pairs can be ranked. The relation is a transitive relation if whenever and then . For any element , the corresponding indifference curve, is made up of all elements of which are indifferent to . Formally, . Formal link to utility theory In the example above, an element of the set is made of two numbers: The number of apples, call it and the number of bananas, call it In utility theory, the utility function of an agent is a function that ranks all pairs of consumption bundles by order of preference (completeness) such that any set of three or more bundles forms a transitive relation. This means that for each bundle there is a unique relation, , representing the utility (satisfaction) relation associated with