am_lect_16 - Another proof involving composition If f : A B...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Another proof involving composition If f : A B is onto and g : B C is onto then g o f : A C is onto Proof : Let y C . Since g is onto, z B such that g ( z ) = y Also since f is onto, x A such that f ( x ) = z Now g o f ( x ) = g ( f ( x )) = g ( z ) = y Hence, x A such that g o f ( x ) = y and hence g o f is onto Are the converse statements true? — If g o f : A C is 1-to-1, is f : A B 1-to-1 and g : B C 1-to-1? — If g o f : A C is onto, is f : A B onto and g : B C onto? (No, for both questions!)
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Without loss of generality Recall that we sometimes do proofs by cases . Consider an example where two cases are very similar: Function f : A B is increasing if x A , y A , x < y f ( x ) < f ( y ) Claim : Any increasing function is one-to-one Proof : We need to show that x A , y A , f ( x ) = f( y ) x = y or equivalently, x A , y A , x y
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/24/2010 for the course CS CS 173 taught by Professor Fleck during the Spring '10 term at University of Illinois, Urbana Champaign.

Page1 / 4

am_lect_16 - Another proof involving composition If f : A B...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online