am_lect_15 - R + Z + is defined as o ( x ) = ( x ) Note...

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Onto vs. One-to-one functions f : A → B is onto if x B , y A , f ( y ) = x f : A → B is one-to-one if x A , y A , f ( y ) = f ( x ) x = y Examples :   : R Z is onto because x Z , y = x R , y = x : R + R + is one-to-one because x , y R + , x = y ( x ) 2 = ( y ) 2 x = y : R + R + is also onto because x R + , y = x 2 R + , y = x — Thus : R + R + is bijective Note that   : R Z is not one-to-one because 2.5 = 2 but 2.5 2
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Composing two functions Suppose f : A B and g : B C . Then g o f : A C is defined as g o f ( x ) = g ( f ( x )) Example : Consider : R + R + and   : R + Z + . Then o   :
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Unformatted text preview: R + Z + is defined as o ( x ) = ( x ) Note that order of composition matters: f o g g o f in general! Claim: If f : A B and g : B C are both one-to-one, then g o f : A C is also one-to-one Proof : Suppose x A , y A such that g o f ( x ) = g o f ( y ) By definition, g ( f ( x )) = g ( f ( y )) Since g is one-to-one, this means that f ( x ) = f ( y ) Since f is one-to-one, this means that x = y , which completes the proof...
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am_lect_15 - R + Z + is defined as o ( x ) = ( x ) Note...

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