📊SET THEORY

  ELEMENTARY SET THEORY

A Set is a well-defined collection of objects or things. e.g.   P= {Prime numbers from 1-30}

Elements of a Set: are the items/things that belong to the set. Its denoted by £   e.g.
         Let Q = {1,2,3,4,5}

Since 2 is contained in set Q, then 2£Q.

But 7 is not  contained, therefore, 7 is not a member of set Q

Number of elements in a set; is also called _CARDINAL NUMBER_ of the set. It is written as n(A).
where n= number of
            A= Set
Hence,
Given A= {2,4,6,8}
       :- n(A) = 4.

That's, number of elements in set A is 4✅✅✅
  On this same note:

If P= {2,2,2,7,8,8,8,8}.  Find n(P).

Solution
If you  have numbers or elements being repeated, you pick it as one

:- n(P) = 3✅✅✅*

Subset: is a set within another set. 

Assume X= {3,5,6,7,8,9,10}
                Y= {5,7,8}

Y is said to be a subset of X because elements of set Y are contained in set X.
Mathematically: Y C X

*Power set:* is the number of subsets a particular set have. This can easily be determined.
Consider,
       If X= {2}
subsets are {2} and { }

Power set = 2ⁿ⁽ˣ⁾
                    = 2¹    = 2✅✅✅
  On the same note;

If R = {4,6,7}
Subsets are {4}, {6}, {7}, {4,6}, {4,7}, {4,6,7} or R, { }
    
Power set = 2³    = 8✅✅✅

*Let's quickly do some exercises*
  *QUESTIONS*
*A*
Find the number of subsets in the following sets:
1️⃣. T = {a, b, c, d}
2️⃣.  S = {Even numbers less than 12}

*Show me your answers*

*ALGEBRAIC SET NOTATION*
  D is a set of x such that x is an integer and lies between 5 and 14 both inclusive. i.e x greater or equal to 4, x less or equal to 14.
  This means:

Mathematically;.

     *D= {x : x £ Z5 ≤ x ≤ 14}*

Which implies  that;

     *D= {5, 6, 7, 8, 9, 10, 11, 12,13, 14}*
  *Note*

❓ *The elements of a set must be separated from one another with commas*

❓ *Curly bracket is the only acceptable bracket for set*

*TYPES OF SET*
  1️⃣. *Empty set:* is a set without any element or member. Also known as *NULL or VOID set*

Its denoted by { } or ∅
  *Note:*
❓❓❓
*{O} is not an empty set because it contains element Zero*
 
2️⃣. *Finite set:* Elements that can
be counted. i.e the counting of the elements has a definite end.

*Examples:*
1. {Students in Nigerian Universities}
2. {Number of grains in a bag of beans}

No doubt, this is countable just that the time to complete it may be long.

3️⃣. *Infinite set:* Elements are uncountable.

*Examples*
1. {Multiples of 9}
2. {Positive integers}

4️⃣. *Disjoint set:* Two sets are disjoint if they have no element in common.

*Example*
    F= {a, b, c}
    G= {d, e, f}

Then F and G are disjoint

5️⃣. *Equal set:* If they contain the same members. The order or arrangement of the members doesn't matter

*Example*
If A= {1, 3, 4, 7}.  
    B= {3, 7, 1, 4}

Hence A=B

*Let's do some exercises*🔥🔥🔥
*B*
*QUESTIONS*
1️⃣. List the members of the following sets:

(a). {Prime numbers less than 40}

(b). {Factors of 12 greater than 3}
*C*
*QUESTIONS*
2️⃣. Find the membership of the following sets:

(a) P= {x : x £N, x < 13}

(b). Q= {y: y £ integers, -4 < y ≤ 9}

*D*
*QUESTION*

3️⃣. Find the Cardinal number of;
      *X= {x : x ≤ 7, x £ N}*

*E*
*QUESTION*

4️⃣. Find the power set of set:
      *F= {y: y £ N, 6> y}*

*RELATIONSHIP BETWEEN SETS*
  1️⃣. *Intersection of set:*  it's denoted as PnQ which means elements common to sets P and Q

*Example*
1. If P= {1,2,3,4}
and. Q= {4,8,2,12}

*PnQ = {2,4}*

2️⃣. *Union of sets:*  it is denoted as PuQ which means combining the elements of the two or more set in one common set without replacement (repeating any element).

*Example*
If A= {1,3,5,7,9,11}.   And
    B= {1,2,3,4,5,6}

*AuB = {1,2,3,4,5,6,7,9,11}*

3️⃣. *The Universal set:* it is denoted by ∪ (in another ream😹). Or sometimes E.
Is the background set which contains all elements being discussed under each smaller set(subset)

4️⃣. *Compliment of a set* Compliment of set P is P' or P^c. It is set of all elements which are present in the universal set but are not in set P itself.

*Example*
U= {1,2,3,...,8,9,10}

P= {2,4,6,8}

Therefore:

*P'= {1,3,5,7,9,10}*

I believe those definitions are simple...

*Let's have some exercise*
*F*
*QUESTIONS*

1️⃣. If U= {1,2,4,6,9,12}
            A= {4,6,12}
            B= {2,4,9}

Find;
(a). A'
(b).  B'
(c).  AnB
(d). (AuB)'

*DIFFERENCE OF SETS*   Given A and B, the difference is denoted by 'A-B' which means set of elements in A but not in B or vice-versa

*Example*
If A= {a,b,c,d,e,f}
    B= {b,d,e,g,h}

*A-B = {a,c,f}*
*B-A= {g,h}*

*G*
* QUESTION*

*A is a set of x such that x is an integer and lies between 5 and 14 both inclusive. i.e x greater or equal to 4, x less or equal to 14.*

Write this mathematically❗❗❗
 

*H*
*QUESTION 7️⃣*

Write out the members of the following sets:

(a). *{x : x £N, x ≤ 10}*

(b). *{p : p is a factor of 12}*

(c). *{q: q £N, 10 < q < 22}*

*The brings us to the end of today's 1st tutorial🟠🔴🔵*

Drop your answers as comment.