General Mathematics
General Mathematics consists of many tools of mathematics which are extensively used at many places in many chapters of mathematics.
These topics of general mathematics is very important as without understanding of these, students face lot of difficulties in understanding different chapters of mathematics.
The topics covered in General mathematics are:-
* Graph of some standard functions and transformation of graph.
Sets
One of most important concept of General mathematics is Understanding of Sets.
What is a Set ?
I am sure this question must be going on in your mind.
In very simple and understandable way we can call the grouping of objects in a group.
e.g (a) Group of students in your class is a set.
(b) Group of people living at your home is another set.
Remember in both these group you are one of the member. You are member of two different sets.
With this understanding now lets define a set.
A Set is a well-defined collection of objects.
Sets are usually denoted by capital letters e.g A, B, X, Y, Z,…. and elements are generally denoted by small letters e.g a, b, c,….
If X is a set and x is the element of a set then we say $ x \in X $ and is read as x belongs to X. The symbol $\in $ is read as belongs to.
Also if we want to write element a doesn’t belongs to set A. It is written as $ a\notin A $.
Illustration 1. Which of the following are Sets?
(a) The collection of integers between 0 and 10 both inclusive.
(b) The collection of capital alphabets from English letters.
(c) The collection of best 11 players for cricket team.
(d) The collection of best questions for practice.
Answer. (a) The collection of integers between 0 and 10 both inclusive, is a Set because the criteria is well defined for the formation of group. i.e X = {x| $x\in I$ and $0\leq x \leq 10$}.
(b) The collection of capital alphabets from English letters is a Set as the objects are well defined are Capital alphabets of English letters.
A = {A, B, C,….,Z}
(c) The collection of best 11 players for cricket team, is not a Set as the property of selection of players is not well defined. Best is a relative term.
(d) The collection of best questions for practice is not a Set. As in this the Best is relative term, does not well define the selection.
Different Ways of writing a Set.
There are two ways in which a set can be written.
- Roster Form.
- Set Builder Form.
Roster Form.
Roster form is with in curly brackets we write the elements of set separated by commas.
e.g Set X = {1, 2, 3, 4 }
Set Y = { a, b, c, d }
Remember.
- In a set none of the element be repeated i.e each distinct element will exist exactly once in a set. like if collection of objects are 0, 1, 1, 2, a, a, b then the set consisting of these elements will be { 0, 1, 2, a, b} only.
- Elements can be written in any order in a set. i.e Let us consider set X = {1, 2, 3, 4 } can also be written as X = { 3, 1, 4, 2 } or can be written as X = { 1, 4, 3, 2}
- An infinite set can be written by writing dots after some of elements. Like Set of natural numbers can be written as Set N = { 1, 2, 3, …. }
Remember.
1). In a set none of the element be repeated i.e each distinct element will exist exactly once in a set. like if collection of objects are 0, 1, 1, 2, a, a, b then the set consisting of these elements will be { 0, 1, 2, a, b} only.
2). Elements can be written in any order in a set. i.e Let us consider set X = {1, 2, 3, 4 } can also be written as X = { 3, 1, 4, 2 } or can be written as X = { 1, 4, 3, 2}
3). An infinite set can be written by writing dots after some of elements. Like Set of natural numbers can be written as Set N = { 1, 2, 3, …. }
Set Builder Form.
In Set Builder form a set is defined by describing the properties of its elements. We do not write the elements directly in a set builder form.
e.g Set of natural numbers less than or equal to 4 can be written in set builder form as X = { x | $x \in N ,\; x \leq 4 $ }
Set of all letters of English lower case alphabet Y = { x| x is lower case English alphabet }
Set Builder form is mainly used to describe the infinite set or sets having large number of elements.
- Set of natural numbers Set A = { x | $x \in N$}
- Set of first 1000 natural numbers Set B = { x | $x \in N \;,\,x \leq 1000 $ }
* Set of natural numbers Set A = { x | $x \in N$}
* Set of first 1000 natural numbers Set B = { x | $x \in N \;,\,x \leq 1000 $ }
Illustration 2. Write the following Sets in roster form.
(a) Solutions of equation (x-1)(x-2)(x-5)(x-10)=0
(b) Collection of natural number multiple of 3 and less than and equal to 18.
Answer. (a) Solutions of equation (x-1)(x-2)(x-5)(x-10)=0 are x=1, 2, 5, 10.
Hence required Set is X = {1, 2, 5, 10 }
(b) Natural number multiple of 3 and less than and equal to 18 are 3, 6, 9, 12, 15, 18.
Hence required set is Y= {3, 6, 9, 12, 15, 18}
Illustration 3. Write the following sets in set builder form.
(a) A=$\{\cfrac{1}{2}, \cfrac{1}{5},\cfrac{1}{10},\cfrac{1}{17}, \cfrac{1}{26},\cfrac{1}{37}\}$
(b) B={4, 7, 10, 13, 16, 19, 22}
Answer. (a) The general term in the set is $t_r=\cfrac{1}{r^2+1}$ where r is a natural number and is less than or equal to 6.
Hence in set builder form is A = {x | $x=\cfrac{1}{r^2+1}\,,r\in N\,, r\leq 6$}.
(b) The general term is $t_r=3r+1$. where r is a natural number less than or equal to 7.
Hence in set builder form is B = { x | x = 3r + 1 , $r\in N$ and $r\leq 7$}.
Types Of Sets.
The two types of sets are UNIVERSAL SET and EMPTY SET.
UNIVERSAL SET .
A set which consist of all elements under study / consideration is called as UNIVERSAL SET. The standard notation used to represent Universal set is U.
Universal set is question specific, or is specific to all elements / objects under discussion or study.
e.g If the question is related to integers, the universal set can be set of all integers. If the study is all rational numbers then universal set could be set of all rational numbers.
EMPTY SET .
Let us consider a set of natural numbers less than zero.
i.e Set X={ x | $x \in N\;,\; x \lt 0 $ }.
Set of real numbers less than or equal to -2 and greater than or equal to 5.
Set Y = { x | $x \leq -2\;,\; x\geq 5$ }
These sets do not have any element in the set. The sets which does not have any element in them is called as EMPTY SET. Empty set is denoted by $ \varnothing $.
What is Cardinal number of a set ?
The number of elements in a set is called as the cardinal number of a set. Cardinal number of a set is always a whole number. If X is a set consisting of first 4 prime numbers i.e Set X = { 2, 3, 5, 7}. The Cardinal Number of set X is denoted by n(X) and for given set X number elements are four hence cardinal number of set x is 4. i.e n(X)=4 .
The Cardinal number of an Empty set is zero. $n(\varnothing)$ = 0.
Based on the number of elements in a set the set is classified as FINITE SET or INFINITE SET.
EQUAL SETS
Two sets are equal if they both have equal cardinal number and each element of one set belong to second set also.
e.g Set X = { 2, 3, 5, 7 } and Set Y = { 3, 7, 2, 5 }
Now we can see n(X) = n(Y) = 4 and every element $x\in X $ also belong to set Y. Hence Set X and Set Y are two equal sets.
SUBSET.
A set X is called as subset of set Y if all elements that belongs to set X also belongs to set Y. If $ \forall x \in X $ if $ x\in Y $ then X is a subset of Y and is denoted as $X\subseteq Y $
If there exist at least one element in set X that doesn’t belongs to set Y then set X is not the subset of set Y and is denoted as $ X \not\subset Y $
- Set itself is always a subset of itself.
- Empty set is subset of every set.
If Set A is subset of Set B ( $ A \subseteq B $ ) and Set B is subset of set A ( $B \subseteq A $ ) then two sets are equal sets ( $ A = B $ ).
SUPER SET.
If set X is a subset of set Y then Y is called as super set of set X. Is denoted as $Y \supseteq X $
Universal set is super set of all sets under consideration.
POWER SET.
The collection of all subset of a given set A is called as power set of A, and is denoted as P(A). If n(A)= n than n[P(A)]=$ 2^n $.
Power set include every subset i.e empty set and set itself also.
e.g Let A be a set A = {a, b, c}
Then all possible subsets of set A are $\varnothing $, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}.
Now Power set of set A is
P(A) = { $\varnothing $, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c} }
Hence Cardinal number of Power Set of Set A is n[P(A)]=$2^3$ = 8
VENN DIAGRAMS.
In order to visualize different operations / relations between the sets, the sets are represented by means of a diagram called as Venn Diagrams. The Universal set is represented by a rectangle and its subsets are represented by the circles.
The elements of the set are written within the circle representing that set.
The Universal set U = {1, 2, 3, 4, 5 } and set A = {1, 4, 5 }
Set Operations
Since Sets are consisting of objects like numbers and variables, Hence certain required operations can be performed with sets. Like combining of sets, taking common between sets, subtracting the sets are some General Mathematic operation. These operations are also called as Union of sets, Intersection of sets, Complement of a set, Subtraction of sets.
Union Of Sets.
Union of two sets A and B is defined as the collection of objects that belong to either Set A or Set B also to both Sets A and B. Symbol used to represent Union of sets is $ \cup $. Union of set A and B is represented as $A \cup B$.
$ A \cup B $ = { x | x $\in A $ or x $\in B$ }
Let Set A = { 1, 2, 3, 4 } and Set B = { 2, 4, 6, 8, 10,12,14 }
Then Union of Sets will be
($A \cup B$) = { 1, 2, 3, 4, 6, 8, 10, 12, 14 }
Remember
- Union of any set with Empty Set ( Null Set ) will always be the set itself. i.e A $\cup\, \varnothing$ = A.
- Union of a set with universal set will always be a universal set. i.e A $\cup$ U = U.
- If A is a subset of set B or B is a super set of set A then Union of two sets will always will be set B. i.e if A $\subseteq$ B or B $\supseteq$ A then A $\cup$ B = B.
- If A $\cup$ B = B then A is a subset of B or B is super set of A.
Remember
1). Union of any set with Empty Set ( Null Set ) will always be the set itself. i.e A $\cup\, \varnothing$ = A.
2). Union of a set with universal set will always be a universal set. i.e A $\cup$ U = U.
3). If A is a subset of set B or B is a super set of set A then Union of two sets will always will be set B. i.e if A $\subseteq$ B or B $\supseteq$ A then A $\cup$ B = B.
4). If A $\cup$ B = B then A is a subset of B or B is super set of A.
Intersection Of Sets.
Intersection of two sets A and B is collection of common objects that belongs to both the sets. Symbol used to denote intersection of two sets $\cap $ between two sets. Intersection of Set A and Set B is represented as $A \cap B$.
Intersection of Set A and Set B is
A$\cap$ B = { x | x $\in$ A and x $\in$ B }.
Like if Set A = { 1, 2, 3, 4 } and Set B = { 2, 4, 6, 8, 10, 12, 14 } then
A$\cap$B = { 2, 4 } Since 2$\in$ A and 2$\in$ B, Similarly element 4 also belong to both.
Remember
- Intersection of any set with Empty set ( Null Set ) will always be Empty set ( Null set ). i.e A $\cap\,\varnothing$ = $\varnothing $.
- Intersection of any set A with universal set will always be set A itself. i.e A $\cap$ U = A.
- If A intersection B is set A then A is subset of set B or set B is super set of set A. i.e if A $\cap$ B = A then A $\subseteq$ B or B $\supseteq$ A.
- If intersection of two sets is a null set then the sets are called as disjoint sets. $A\cap B\,=\,\varnothing$ then Set A and Set B is called DISJOINT SETS
Remember
1). Intersection of any set with Empty set ( Null Set ) will always be Empty set ( Null set ). i.e A $\cap\,\varnothing$ = $\varnothing $.
2). Intersection of any set A with universal set will always be set A itself. i.e A $\cap$ U = A.
3). If A intersection B is set A then A is subset of set B or set B is super set of set A. i.e if A $\cap$ B = A then A $\subseteq$ B or B $\supseteq$ A.
4). If intersection of two sets is a null set then the sets are called as disjoint sets. $A\cap B\,=\,\varnothing$ then Set A and Set B is called DISJOINT SETS
Difference Of Sets.
Difference of Set B from Set A i.e A – B is a set that consist of objects of set A that are not present in B. If an element a $\in$ (A-B) then a $\in$ A but a $\notin$ B.
i.e (A-B) = { x | x $\in$ A and x $\notin$ B }
It can also be put as from first set eliminate the common elements of both sets.
A – B = A – (A $\cap$ B )
e.g Let two sets are Set A= { 1, 2, 3, 4 } and Set B = { 2, 4, 6, 8, 10, 12, 14 }
than Set ( A – B ) = { 1, 3 }
Remember
- A – $\varnothing$ = A
- If A – B = $\varnothing$ then A is a subset of set B
- If A – B = A then A and B are Disjoint Sets (i.e $A\cap B\,=\,\varnothing$ ).
Symmetric Difference of Sets.
Symmetric difference of two sets A and B is collection of objects that belongs to either A or to B but does not belongs to both. Symbol used is $\Delta$ between two sets. i.e $A\Delta B$ is symmetric difference of set A and set B.
$A\Delta B$ = { x | $x\in A$ or $x\in B$ but $x\notin (A\cap B)$ }.
can also be written as $A\Delta B$ = $(A\cup B)$ – $(A\cap B)$.
Complement Of a Set.
Take a difference of set A from Universal set the set we get is complement of a set. i.e U – A = $A^c$ where $A^c$ is read as complement of set A.
Hence $A^c$ = { x | x $\in$ U and x $\notin$ A }.
Remember
- $A \cup A^c$ = U —> is universal set.
- $A \cap A^c\,=\,\varnothing$ —> A and $A^c$ are disjoint sets.
- A – $A^c$ = A and $A^c$ – A = $A^c$
Principle Of Inclusion Exclusion.
Cardinal number of elements in Union and Difference of Sets.
Cardinal Number of Union of two sets A and Set B.
Total number of elements in set A or B will be add number of elements in A and in B, in this the common to both gets added with both, they are substracted once.
$n(A\cup B)$ = n(A) + n(B) – $n(A\cap B)$
Similarly Cardinal number of elements in Union of three sets say A, B and C.
$n(A\cup B\cup C)$ = n(A) + n(B) + n(C) – $n(A\cap B)$ – $n(B\cap C)$ – $n(A\cap C)$ + $n(A\cap B\cap C)$.
Cardinal number of Difference of set.
n(A – B) = n(A) – $n(A\cap B)$
Order Pair
If a and b be two elements such that a$\in$A and b$\in$B (where A and B may be same or distinct Sets), then order pair of a and b is denoted as (a, b), where a is called as first component and b is called second component. Interchanging the position of two components in a pair will give us different order pair.
i.e order pair (a, b) $\neq$ (b, a)
Two order pair (a, b) and (c, d) are equal (or same) if and only if a=c and b=d.
Cartesian Product of two Sets.
Cartesian product of two sets A and B is the set of all order pairs where first member belongs to set A and second member belongs to set B, And is denoted as A x B read as “A cross B”.
i.e Let Set A={a, b, c} and B={1, 2, 3} then cartesian product of two set is A x B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3), (c, 1), (c, 2), (c, 3)}
also can be written as A x B = {(x, y) : x$\in$A and y$\in$B }
Example 1 :: In a survey of 425 students in a school, it was found that 115 drink apple juice. 160 drink orange juice and 80 drink both apple as well as orange juice. How many drink neither apple juice orange juice?
Solution 1 :: Let A be students who drink Apple i.e n(A)= 115 and B be students those who drink Orange i.e n(B)=160.
n(A$\cap$ B) = 80
We know that Number of students drinking at least one of Apple or Orange juice is n(A$\cup$ B) = n(A) + n(B) – n(A $\cap$ B).
$\Rightarrow$ n(A$\cup$ B) = 115 + 160 – 80 =195
Number of Students who neither drink Apple juice or Orange juice are (A$\cup$ B)^c = n(U) – n(A$\cup$ B)
Where n(U)= 425 Total number of students surveyed.
$\therefore$ (A$\cup$ B) = 425 – 195 = 230.
Hence from 425 students surveyed 230 drink neither Apple juice nor Orange juice.
Example 2 :: If the difference in number of subsets of two sets is 112 than find the number of elements in each set.
Solution 2 :: Let A and B be two sets such that n(A)=n and n(B)=m also assuming n>m.
Hence number of subsets of A and B are $2^n$ and $2^m$.
$\therefore \; 2^n – 2^m=112 $
Hence $2^m(2^{(n-m)}-1)=2^4\times 7$
$\therefore \; 2^m=2^4\;and\; 2^{(n-m)}-1=7$
Hence m=4 and n-m=3
$\therefore $ m=4 and n = 7.
Hence number of elements in each sets are 7 and 4.
Example 3 :: Let $A_1, A_2,A_3,…,A_{30}$ be thirty sets each with five elements and $B_1, B_2,B_3,…,B_n$ are n sets each with three elements.
If $\bigcup_{i=1}^{30}A_i= \bigcup_{j=1}^nB_j=S$
is such that each element of set S belongs to exactly 10 of the $A_i’s$, and to exactly nine of $B_j’s$. Find n?
Solution 3 :: Let $A_1, A_2,A_3,…,A_{30}$ be thirty sets each with five elements and $B_1, B_2,B_3,…,B_n$ are n sets each with three elements.
and
$S= \bigcup_{i=1}^{30}A_i= \bigcup_{j=1}^nB_j$
Let n(S)=m and given that every distinct element of Set S belongs to exactly 10 sets of $A_i’s$
$\Rightarrow\, \sum_{i=1}^{30} n(A_i)=10m$
$\therefore\;10m=150$
Hence m=15
And $\sum_{j=1}^n=3n=9m$
Hence n=3m=45.