13397847
Description
Mind Map by Luke Byrne, updated more than 1 year ago
|
|
Created by Luke Byrne
about 6 years ago
|
|
Cartesian Products
- To understand sets like R^1 in a more theoretical way.
- From calculus...
- R = R^1 ∋ x
- R x R = R^2 ∋ (x1, x1)
- ...
- R x ··· x R = R^n ∋ (x1, x2, ..., xn)
- Cartesian Products
- R = R^1 ∋ x
- Let A, B be sets
- The Cartesian product,
denoted by A x B, consists
of all ordered pairs (x, y),
s.t. x ∈ A and y ∈ B.
- A x B = {(x, y) | x ∈ A ∧ y ∈ B}
- A x B = {(x, y) | x ∈ A ∧ y ∈ B}
- The Cartesian product,
denoted by A x B, consists
of all ordered pairs (x, y),
s.t. x ∈ A and y ∈ B.
- A = {1, 3, 7}
- B = {1, 5}
- Cartesian Product
- A x B = {(1, 1), (1, 5), (3, 1), (3, 5), (7, 1), (7, 5)}
- The order matters: (7, 1) is different
from (1, 7). This is why (x, y) is called an
ordered pair.
- The order matters: (7, 1) is different
from (1, 7). This is why (x, y) is called an
ordered pair.
- A x B = {(1, 1), (1, 5), (3, 1), (3, 5), (7, 1), (7, 5)}
- A = {(x, y) ∈ R^2 | x^2 + y^2 = 1}
- circle of radius 1
- circle of radius 1
- B = {z ∈ R | -2 ≤ z ≤ 2} = {-2, 2}
- closed interval
- closed interval
- Cartesian Product
- A x B ← cylinder of radius 1 and height 4
- A x B ← cylinder of radius 1 and height 4
- Cardinality in a Cartesian product
- A has n elements
- B has p elements
- Cartesian product (A x B) has np elements
- A has n elements
Want to create your own Mind Maps for free with GoConqr? Learn more.