An infinitely large napkin Sets and functions

14 flashcards · Mathematics · by matthew Shing

1. what does ∈ mean

   We write x ∈ S to mean “x is in S”, for example 3 ∈ N

2. what does ∪ mean and what is it called

   ∪ B for the set of elements in either A or B (possibly both), called the union of A and B.

3. what does ∩ mean and what is it called

   A ∩ B for the set of elements in both A and B, and called the intersection of A and B.

4. what does this set mean

   C is the set of complex numbers, like 3.2 + √2i

5. what is this set

   R is the set of real numbers, like √2 or π.

6. What is this set

   N is the set of positive integers, like 5 or 9

7. What is this

   Q is the set of rational numbers, like 7/3.

8. What is this set

   Z is the set of integers, like −2 or 8.

9. Given two sets A and B, What does a function mean in that context

   f from A to B is a mapping of every element of A to some element of B

10. Define injective Functions

    A function f : A → B is injective if it is “one-to-one” in the following sense: if f (a) = f (a′) then a = a′. In other words, for any b ∈ B, there is at most one a ∈ A such that f (a) = b. Often, we will write f : A ↪→ B to emphasize this. There’s a function taking every human to their age in years (rounded to the nearest integer). This function is not injective, because for example there are many people with age 20. This function is also not surjective: no one has age 10000.

11. Define Surjective Functions

    A function f : A → B is surjective if it is “onto” in the following sense: for any b ∈ B there is at least one a ∈ A such that f (a) = b. Often, we will write f : A  B to emphasize this. There’s a function taking every USA citizen to their social security number. This is also not surjective (no one has SSN equal to 3), but at least it is injective(no two people have the same SSN).

12. Define bijective Functions

    A function f : A → B is bijective if it is both injective and surjective. In other words, for each b ∈ B, there is exactly one a ∈ A such that f (a) = b. In a classroom with 30 seats, there is exactly one student in every seat. Thus the function taking each student to the seat they’re in is a bijection; in particular, there are exactly 30 students

13. If set A is at most the size of set B what is A->B called

    Injective Assume for convenience that A and B are finite sets. Then: ˆ If f : A ↪B is injective, then the size of A is at most the size of B. ˆ If f : A →→B is surjective, then the size of A is at least the size of B. ˆ If f : A → B is a bijection, then the size of A equals the size of B.

14. define ∼ Equivalence relations

    ∼ on X is a binary relation that satisfies the following conditions Reflexive: we have x ∼ x. ˆ Symmetric: if x ∼ y then y ∼ x ˆ Transitive: if x ∼ y and y ∼ z then x ∼ z. An equivalence class is then a set of all things equivalent to each other. One can show that X becomes partitioned by these equivalence classes Let N denote the set of positive integers. Then suppose we declare a ∼ b if a and b have the same last digit, for example 131 ∼ 211, 45 ∼ 125, and so on. Then ∼ is an equivalence rela…