How do you write the composition of the functions and where , , and are sets?
What does mean given and are sets?
The represents the set of functions . This set can be represented as:
Given some set , what is its identity function?
Given some set , what does mean?
That for all , the function . Basically, the function returns any element it is given of some set unchanged. An example being: .
Definition 2.1.2.8 (Isomorphism)
Let and be two sets.
A function is an isomorphism if there exists a function such that and . An isomorphism is denoted as where stands for congruency. is invertible and is the inverse of .
If there is an isomorphism , and are isomorphic sets and can be denoted as
Exercise 2.1.2.2 Here is a simplified account of how the brain receives light. The eye contains about 100 million photoreceptor () cells. Each connects to a retinal ganglion () cell. No cell connects to two different cells, but usually many cells can attach to a single cell. Let denote the set of photoreceptor cells and let denote the set of retinal ganglion cells.
According to the above account, does the connection pattern constitute a function , a function or neither one?
Which can also be written as: where
Would you guess that the connection pattern that exists between other areas of the brain are "function-like"?
I would guess that other parts of the brain are function-like!
Exercise 2.1.2.4 Let be the function that sends every natural number to its square, e.g. . First fill in the blanks below, then answer a question.
Answer:
Answer:
Answer:
Undefined
Answer:
Consider the symbol and the symbol .
What is the difference between how these two symbols are used in this book?
means a general rule for a mapping elements between sets whereas specifically denotes what element maps to what in differing sets.
Exercise 2.1.2.5 If is represented by the diagram below, write its image, as a set.
Exercise 2.1.2.6 Let and .
How many elements does have?
Answer: or
How many elements does have?
Answer: or
Exercise 2.1.2.7.
Find a set such that for all sets there is exactly one element in .
Hint: draw a picture of proposed 's and 's.
Answer:
Find a set such that for all sets there is exactly one element in .
Answer: