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Math-Sets
| Question | Answer |
|---|---|
| Informal definition | A set is a collection of mathematical objects, with the collection treated as a single mathematical object. (This is circular, what is a collection?) |
| Familiar sets | Real numbers, complex numbers, integers, empty set |
| ∅ | Empty set |
| {} | Indicates a set |
| Ordering within braces in a set | Doesn't matter in sets |
| Elements presence in sets | An element either is in or not in a set. Multiple listings of the same element don't matter, and elements can't be in a set more than once. |
| ∈ | Denotes an element is a member of a set. x ∈ A. |
| ∉ | The item is not an element of the set. x ∉ A |
| Membership synonyms | x ∈ A, x is in A, and x is an element of A all mean x is a member of a. |
| Can a set contain sets? | Yes |
| ⊆ | Subset symbol. A ⊆ B |
| Subset-Synonym | Contained in |
| Subset | A ⊆ B means every element of A is also an element of B. ∀x [x ∈ A IMPLIES x ∈ B] |
| Properties of subsets | 1. A ⊆ A 2. ∅ ⊆ everything |
| ∅ ⊆ everything proof | ∅ ⊆ B is defined to mean ∀x [x ∈ ∅ IMPLIES x ∈ B]. x is not part of the empty set so x ∈ ∅ is false. That means the whole implication is true, |
| Defining a set | Generally define a large set by a property in a set. The set of elements x in A such that P(x) is true. {x ∈ A | P(x)} |
| | (pipe) | Such that |
| Union | The set of points in A OR B |
| ∪ | Union |
| Union-Set notation definition | A ∪ B ::= {x | x ∈ A OR x ∈ B} |
| Intersection | Points in both A AND B |
| Intersection-Set notation definition | A ⋂ B ::= {x|x ∈ A AND x ∈ B} |
| Distributive law for Union over intersection | A ∪ (B ⋂ C) = (A ∪ B) ⋂ (A ∪ C) |
| Proving Distributive law for Union over intersection | 1. Prove the two have the same elements- x ∈ A ∪ (B ⋂ C) iff x ∈ (A ∪ B) ⋂ (A ∪ C) for all x 2. x ∈ A ∪ (B ⋂ C) iff x ∈ A OR x ∈ (B ∪ C) (def of ∪) iff 3. x ∈ A OR (x ∈ B AND x ∈ C) (def ⋂) iff 4. (x ∈ A OR x ∈ B) AND (x ∈ A OR x ∈ C) (by distribution o |
| Difference | In one set but not in the other set. |
| − (A – B) | Difference symbol |
| Difference definition-Set notation | A–B ::= {x|x ∈ A AND x ∉ B} |
| Complement-Set notation definition | ‾A (line directly over A) ::= Domain – A = {x|x ∉ A} |
| ‾ A (line directly over A) | Complement of A. |
| . (Period) In set notation | Used for scoping. Substitute for parentheses indicating that the scope of the preceding modifier extends until a parentheses. |
| | | | The number of elements of a set. (e.g. |a| = the number of elements in a) |
| Bit-strings in subsets | Use 1s and 0s to indicate whether an element from a set is present or not. The Kth element of the bit-string is 1 iff a¥k is in the set |
| Power sets and bit strings | 1. # of bit strings = |pow(A)| 2. Corollary: |pow(A)|=2^(|A|) |
| # n-bit strings | 2ⁿ |
| Set size-Properties | 1. |A|=|B|=|C| IMPLIES |A|=|C| 2. |A|>=|B|>=|C| IMPLIES |A|>=|C| 3. |A|>=|B|>=|A| IMPLIES |A|=|B| |
| Well ordered set | A set of W real numbers is well ordered iff it has no infinitely decreasing sequence |
| Superscript asterisk (^*) | If you have some collection of objects which you think of as letters and write ^* that means the finite string of those letters |
Created by:
LuminatedLucy