Give examples of two functions f : N\(\to\) Z and g : Z\(\to\) Z such that g o f is injective but g is not injective.
(Hint: Consider f(x)=x and g (x= IxI )
Give examples of two functions f : N\(\to\) Z and g : Z\(\to\) Z such that g o f is injective but g is not injective.
(Hint: Consider f(x)=x and g (x= IxI )
Solution and Explanation
Define f : N \(\to\) Z as f(x) = x and g : Z \(\to\) Z as g(x) = \(\mid x \mid\) .
We first show that g is not injective.
It can be observed that:
g(-1)=I-1I=1.
g(1) =I1I=1.
∴ g(−1) = g(1), but −1 ≠ 1.
∴ g is not injective.
Now, gof: N \(\to\) Z is defined as gof (x)=g(f(x))=g(x)=IxI.
Let x, y ∈ N such that gof(x) = gof(y).
\(\Rightarrow\) IxI=IyI.
Since x and y ∈ N, both are positive.
Therefore IxI=IyI \(\Rightarrow\) x=y
Hence, gof is injective
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- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
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