Question 1: Find are defined by:
Answer:
Question 2: Let Show that are both defined. Also, find
Answer:
Question 3: Let Show that is defined while is not defined. Also, find
Answer:
Question 4: Let and let and be two functions from to and from to respectively defined as: Show that and both are bijections and find and
Answer:
Question 5: Find when:
Answer:
Question 6: Let be the set of non-negative real numbers. If . Find Are they equal functions.
Answer:
Answer:
Therefore, the domains of are the same.
Answer:
Answer:
So, both have the same domains.
Now
Hence, the associative property has been verified.
Answer:
So, both have the same domains.
Now,
Question 11: Give examples of two functions such that is onto but is not onto.
Answer:
Let us consider given by
Therefore which is an identity function and hence it is onto.
Question 12: Give examples of two functions such that is injective but is not injective.
Answer:
We fist show that is not injective.
It can be observed that
Therefore is not injective.
Hence, is injective.
Question 13: If are one-one functions, show that is a one-one function.
Answer:
Question 14: If
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