Flashcards in Set Theory Deck (20)

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1

## What is a set?

### An unordered collection of objects of same type, called elements/members of set. 2 sets are equal if they contain exactly same elements, no matter the order. Size (cardinality) of set given by card (card ({2,3,5}) = 3. If set contains multiple copies of same element, only counted once. e.g. EvensToTen = {0,2,4,6,8} A set can be empty {}.

2

## How do we show that something is/isn't an element/member of a set?

###
p ∈ EvensToTen if P = 4

p ∉ EvensToTen if P = 3

3

## What are 3 standard sets?

### Z is set of all integers, N is set of all natural nums + R is set of all real nums.

4

## What are 3 ways we can manipulate sets?

### Intersection, union + difference.

5

## What is the symbol for intersection + what does it do?

### A∩B gives elements that are in both sets.

6

## What is the symbol for union + what does it do?

### A∪B gives elements that are in either set.

7

## What is the symbol for set difference + what does it do?

### A\B gives elements in A but not B.

8

## How do we define a more complex set?

### Start with big, easy to define set + throw away values we don't want. Starting set called U (universal) + give conditions for values we want to keep.

9

## Give the notation for defining more complex sets.

###
{x | x ∈ U . x condition }

e.g. x | x ∈ {1,…,10} . x mod 2 = 1} = {1,3,5,7,9}

10

## What are operators for 'and' and 'or' in more complex sets?

###
and = ^ (like intersection)

or = v (like union)

11

## What is the value between { and | known as?

### It is the typical member of the set and can be an expression rather than just a single variable. If every value from universal set is to be used, the condition can be omitted.

12

## Can we use multiple variables in complex set definitions?

### Yes. Just create an expression as normal and use a comma to separate them after the |

13

## How do we create sets where elements are pairs of numbers?

### like this: {[x,y]|x,y ∈ {0,…,10}.x+y=10}

14

## How would we get a set of pairs and remove duplicates?

### {[x,y]|x,y ∈ {0,…,10}.(x+y=10) ^ x<=y}

15

## How would we find the number of pairs which result from a set comprehension?

### card {[d,f]|d∈D,f∈F}

16

## What would we use to say not to include something in a set?

### ≠

17

## How to we add a constraint that needs to be calculated in either order?

### Use abs (absolute value) e.g. abs(age(p1)-age(p2)<2

18

## If everything in set a is in set b, what is it called?

### A is a subset (⊆) of B.

19

## What is a proper subset?

### The sets can't be equal. ⊂. e.g. {1,2} is subset of {1,2,3} but {1,2,3} isn't.

20