Sets

Set theory is the foundation of probability and statistics, as it is for almost every branch of mathematics.

Set is simply a collection of objects; the objects are referred to as elements of the set. The statement that $x$ is an element of set $S$ is written $x\in S$, and the negation that $x$ is not an element of $S$ is written as $x\notin S$. By definition, a set is completely determined by its elements; thus sets $A$ and $B$ are equal if they have the same elements:  

$$A=B\text{if and only if}x\in A⟺x\in B$$


$$\mathrm{If\; A\; and\; B\; are\; sets\; then\; A\; is\; a\; subset\; of\; B\; if\; every\; element\; of\; A\; is\; also\; an\; element\; of\; B:}$$


$$\mathrm{A\; \subseteq \; B if\; and\; only\; if x\; \in \; A\; ⟹\; x\; \in \; B}$$

Every object in a set is unique: The same object cannot be included in the set more than once.

Example 1:

Define each of the following sets by listing its elements:

(a)          The set of the first three months of the year.

(b)          The sets of the whole numbers less than 2.

(c)          The set of letters of the word ‘PAPER’.

Answer:  

(a)          {January,February,March}

(b)          {0,1}

(c)          {P,A,E,R}

   *Note on the Question C, we do not repeat the letter P and the sets has 4 element not 5.*

Example 2:

This question is on subsets.

I   is the set of integers.

N is the set of natural numbers.

Q is the rational numbers

R is the set of real numbers.

(a)       Use the symbol ⊂ to relate I,N,Q,R. 

(b)       Draw a venn diagram to represent I,N,Q and R.

Answer: 

(a)   N⊂I⊂Q⊂R

(b) 

Example 3:

Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

X = {1, 2, 6, 7} and Y = {1, 3, 4, 5, 8}

a) Draw a Venn diagram to illustrate ( X ∪ Y ) ’

b) Find ( X ∪ Y ) ’

Solution:

a) First, fill in the elements for X ∩ Y = {1}

Fill in the other elements for X and Y and for U

Shade the region outside X ∪ Y to indicate (X ∪ Y ) ’

b) We can see from the Venn diagram that

(X ∪ Y ) ’ = {9}

Or we find that X ∪ Y = {1, 2, 3, 4, 5, 6, 7, 8} and so

(X ∪ Y ) ’ = {9} 


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