MAT102 Term Test 2 Prep
Readings 6
06FunctionsII.pdf
- Example 3:
- Determine whether each function below is invertible
- 1
- 2
- Let the transformation be
- doesn't work because of the restriction, won't work for first case
- Exercise 4
- Come up with sets and and a function such that:
- where but
- and then we have a case where we're wondering which created that .
- Surjective, not injective
- where but
- Injective, not surjective
- Exercise 6
- If is a function, is invertible is bijective
-
- Suppose that is bijective, define as follows:
- Fix and consider , this is non-empty as is surjective, and it's element since is injective.
- Call this element
- Define
- Claim is the inverse of
- See that , then
-
- is invertible
- This means by exercise 4 that has and has a .
- To achieve the first case, we need to be surjective, the second, we need to be injective.
- Because we are both, we are bijective.
- Exercise 9
- Show that if then
- Show that there's an injection where
- Injective
- So it's injective
- Because it's injective, this means that
- Example 12
- Show that
- We need to construct an injection both ways, or bijection directly.
- Bijection
- Injection
- Surjection
- Fix an arbitrary
- See that any element from can be created from an element in .
- This is explicitly bijective so it's the same Cardinality.
- Exercise 13
- Show that
- Injection from
- Defined as
- Injective:
- Injection from
- Defined as
- Exercise 19
- Show that
- Injection
- Let
- Injective
- By FTA we have unique prime factorizations.
- Injection
- Let
- By CSB, there is a bijection so they have the same Cardinality
- 1
- Determine whether it's invertible
- a:
- It's not invertible because it's not bijective
- See that many rationals map to
- b:
- Let's see if the transformation will give us the identity
- 2
- A function is left-invertible if there is such that
- Show that is left-invertible is it injective
-
- is left-invertible so there is such that
- Want to show that
- The nature of this necessity means that there shouldn't be a loss of information from
- The identity function will always produce the input
-
- We know that is injective
- Show it's left-invertible
- We want to create some where it creates
- So then
- Since is injective, the outputs are unique, there are no two outputs for one input.
Readings 7
07NumberTheoryI.pdf
- Exercise 2
- For what values of does
- See that for some
- However this is true for all .
- Exercise 4
- if and then
- for
- We know and
- Want to show that
- let
- Proven
- if and then
- for some
- Cases:
- then see that our only two options are or
- if is non-zero and then
- Let as for some
- We can then split them as
- This implies that
- Example 6
- Show that for any that
- for some
- FLT says that
- So this is true
- Directly:
- Case 1:
- is a multiple of
- The product of three consecutive numbers is divisible by
- Case 2:
- which is another multiple of
- Case 3:
- Another multiple of
- Corollary 15
- then
- let
- Corollary 16
- If the integer is a common divisor of both and
- and
- There exists such that then
- and
- Exercise
- 1
- Suppose that
- Prove that if and then
- for some
- Want to show that for some
- Done
- Prove that if then
- and
- Want to show that
- If we multiply them together:
- let
- 2
- Suppose that is odd
- Show that
- So clearly as we have a multiple of .
- 4
- Let with
- Show that for any satisfying there is a such that
- We know that
- Want to show that
- for some
- Let's modify
- Clearly this shows that which means
- All we are doing is changing the variable and sign
- So is the multiplicative inverse
- So then
- 5
- If , show that
- The euclidean algorithm requires that , meaning you can keep on adding or subtracting to .
- Euclidean Algorithm
-
- So
- So
- Lemma:
- Let and
-
Readings 8
08NumberTheoryII.pdf
- Prop 1
- If and then
- for some
- So
- Theorem 4
- If then has a solution
-
- has a solution
- and
-
- Show that has a solution
- for some
- Example 6:
- Find a solution to
-
-
- Theorem 13
- A number is prime for any , if then or
- Exercise 15
- Show that primality is necessary for Theorem 13
- if is not prime, show a counter example
- but or
- but
- Exercise 16
- is prime
-
- By transitivity
-
- By theorem 13
- if is prime, then or
- is prime,
- so
- Theorem 20
- Exercises:
- 2
- How many 's occur at the end of .
- The product of consecutive numbers is divisible by
- Multiples of
- 3
- Let write in terms of its prime factorization
- where
-
- for some
- Suppose that did not have the same prime factorization
- Contradiction because this implies that the primes are not unique.
-
- Since it has the same primes as
- and
- then we have that
- Let
- 4
Readings 9