Set Theory:
Set theory is the branch of mathematics logic that studies sets, which can informally be described as the collections of objects.
$A={\{a,b,c,d,e}\}$
The union of a collection of sets is the set of all elements in the collection.
$A\,\cup{B}={\{x:x{\in}A \,or\, x:x{\in}B \,or\, x{\in}A\, and \,B\,both}\}$
$A_1\,\cup{A_2}\,\cup\,.......,\cup{A}_m={\cup_{i=1}^{m}}\,Ai$
Intersection:
The intersection of two sets A and B is the set of all those elements which are common to both A and B.
$A\,\cap{B}={\{x:x{\in}A \,or\, x:x{\in}B \,or\, x{\in}A\, and \,B\,both}\}$
$A_1\,\cap{A_2}\,\cap\,.......,\cap{A}_m={\cap_{i=1}^{m}}\,Ai$
Complementation:
The complement of set A, denoted as $\bar{A} \, or A^{c}$ is the set of all elements not contained in A.
$A=\{{x:x\notin{A}}\}$
Hence, $\bar{A}=\cup-{A}$
Subset:
Set A is a subset of a set B if all elements of A are also elements of B. Here, B is the then a superset of A.
$\therefore\,A\subseteq{B}$
It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B.
For example: $A\cap{B}\subset{A\cup{B}}$
Difference:
The difference $A-B$ is the set of all elements contained in A but not in B. Similarly, $B-A$ is the set of all elements contained in B but not in A.
$A-B=\{{x:x\in{A}\, but\, x\not\in{B}}\}$
$B-A=\{{x:x\in{B}\, but\, x\not\in{A}}\}$
$\boxed{Note:}$
$A-B=A\cap{\bar{B}}$
$B-A=B\cap{\bar{A}}$
Properties of Union and Intersection:
- Commutativity: $$A\cup{B}=B\cup{A}$$$$A\cap{B}=B\cap{A}$$
- Associativity: $$A\cup({B}\cup{C})=(A\cup{B})\cup{C}$$ $$A\cap({B}\cap{C})=(A\cap{B})\cap{C}$$
- Distributivity: $$A\cap({B}\cup{C})=(A\cap{B})\cup{(A\cap{C})}$$ $$A\cup({B}\cap{C})=(A\cup{B})\cap{(A\cup{C})}$$
- Idempotency: $$A\cup{A}={A}$$$$A\cap{B}={A}$$
They also obey De-Morgan's laws: -
a. $(A\cup{B})^{c}={A}^c\cap{B}^c$
b. $(A\cap{B})^c={A}^c\cup{B}^c$
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Definition of Probability:
The probability is a chance or possibility of happening or occurrence of an event in an experiment.
- The probability is a numeric value that lies between 0 and 1.
- If there is absolute possibility of occurrence of an event then its value is 1. This probability is called certain probability or sure probability.
- If there is complete impossible of happening of an event then its probability is 0. This probability is called probability of uncertainty.
- It is denoted by p or P. $$\therefore{p}=prob.{(E)}$$
Definition of various terms used in probability:
Random Experiment:
The experiment in which outcomes is not unique then it is called a random experiment.
Trial and Event:
Trial is an experiment and getting result from an experiment is called event.
Ex: Tossing of a coin is a trial and getting head or tail is event.
Exhaustive number of cases:
Total number of possible outcomes of an experiment is called exhaustive number of cases.
Ex: In coin toss experiment head and tail are the events, therefore, exhaustive number of cases in coin toss experiment is 2.
⇒Similarly, In dice throw experiment is 6.
Sample Space:
The set of exhaustive number of cases is called sample space. $$S=\{{Head, Tail}\}=\{{H},{T}\}$$
Equally Likely Events:
If the events have the same possibility then the events are said to be equally likely events.
For example;
↠In coin toss experiment, head and tail are equally likely events.
↠In dice throw experiment, numbers of 1,2,...6 are equally likely events.
Favourable Events:
In an experiment, the number of times which entail the happening of an event is called favourable cases.
Dependent Events:
Two or more events are said to be dependent events if one of them affect the occurring of the other in the next (second) trial.
Independent Events:
Two or more events are said to be independent events if one of them not affect the occurring of the other in the second trial.
Mutually Exclusive Events:
Two or more events are said to be mutually exclusive events if the happening of any one events excludes (precludes) the happening of all others in the same experiment / trial.
For example;
↠In coin toss experiment, head and tail are the mutually exclusive events.
↠In dice throw experiment, numbers 1,2,....6 are mutually exclusive events.
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Counting Process:
Permutation and Combination
Permutation:
The number of arrangement of objects in a definite order is called the number of permutation. If out of 'n' objects, 'm' objects are chosen at a time then their number of permutation is denoted by $^np_{r} \, or \, p(n,r)$ and is given by; $$^np_{r}=\frac{n!}{(n-r)!}$$
where $r\le{n}$ and n! is read as n factorial or factorial of n.
↠The permutation of the objects for repeated different:
Let 'n' be the number of objects taken all at a time and let 'p' be the objects of one kind, 'q' be the object of second kind, 'r' be the object of third kind, 's' be the object of fourth kind and so on, then, their number of permutation is given by; $$\frac{n!}{p!\,q!\,r!\,s!\,...}$$
Examples:
🆀There are 10 vacant seats in a bus. How many ways can 4 people be seated?
↠ Here,
Total number of vacant seats(n)=10
The number of seats taken at a time(r)=4
Then,
Number of possible cases = $^np_{r}=\,^{10}p_{4}=\frac{10!}{(10-4)!}=\frac{10!}{6!}=5040$
🆀How many ways letters can be arranged in the word 'STATISTICS'?
↠ Here,
Total number of letters in the word 'STATISTICS' (n)=10
S is repeated three times (p)=3
T is repeated three times (q)=3
I is repeated two times (r)=2
A is repeated one time (s)=1 and
C is repeated one time (t)=1
Then,
Total number of arrangement of letters= $\frac{10!}{3!\,3!\,2!\,1!\,1!}=50400$
Combination:
The number of arrangement of objects without any definite order is called number of combination.
Symbolically,
Let 'n' be the total number of objects and let 'r' be the number of objects taken at a time then the number of combination is denoted by; $^nc_{r} \,or\, C(n,r) $ and is given by;
$$^nc_{r}=\frac{n!}{(n-r)!\,r!}=\frac{{^np_{r}}}{r!}$$
$$\therefore\,\boxed{^np_{r}={r!}\,{^nc_{r}}}$$
Example
🆀There are 12 questions given in an examination. How many ways can 5 questions be chosen?
↠ Here,
Number of questions (n)=12
The number of questions taken at a time (r)=5
Then,
Number of combination = $^nc_{r}=\,^{12}c_{5}=\frac{12!}{7!\times5!}$
=$792\,$ways
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