Every seat had someone sitting there (there were no empty seats), and.Every student was in a seat (there was no one standing),.What the instructor observed in order to reach this conclusion was that: After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. A bunch of students enter the room and the instructor asks them to be seated. In a classroom there are a certain number of seats. Property (3) says that for each position in the order, there is some player batting in that position and property (4) states that two or more players are never batting in the same position in the list. Property (2) is satisfied since no player bats in two (or more) positions in the order. Property (1) is satisfied since each player is somewhere in the list. The set X will be the players on the team (of size nine in the case of baseball) and the set Y will be the positions in the batting order (1st, 2nd, 3rd, etc.) The "pairing" is given by which player is in what position in this order. This symbol is a combination of the two-headed rightwards arrow ( U+21A0 ↠ RIGHTWARDS TWO HEADED ARROW), sometimes used to denote surjections, and the rightwards arrow with a barbed tail ( U+21A3 ↣ RIGHTWARDS ARROW WITH TAIL), sometimes used to denote injections.Įxamples Batting line-up of a baseball or cricket team Ĭonsider the batting line-up of a baseball or cricket team (or any list of all the players of any sports team where every player holds a specific spot in a line-up). īijections are sometimes denoted by a two-headed rightwards arrow with tail ( U+2916 ⤖ RIGHTWARDS TWO-HEADED ARROW WITH TAIL), as in f : X ⤖ Y. With this terminology, a bijection is a function which is both a surjection and an injection, or using other words, a bijection is a function which is both "one-to-one" and "onto". Functions which satisfy property (4) are said to be " one-to-one functions" and are called injections (or injective functions). Functions which satisfy property (3) are said to be " onto Y " and are called surjections (or surjective functions). It is more common to see properties (1) and (2) written as a single statement: Every element of X is paired with exactly one element of Y. Satisfying properties (1) and (2) means that a pairing is a function with domain X. no element of Y may be paired with more than one element of X.each element of Y must be paired with at least one element of X, and.no element of X may be paired with more than one element of Y,.each element of X must be paired with at least one element of Y,.Further information on notation: Function (mathematics) § Notationįor a pairing between X and Y (where Y need not be different from X) to be a bijection, four properties must hold:
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