Sets, Operations on Set

1. Sets, Elements, Operations on Set
Sets Means Collection of Objects(Well Define)(of any type
whatsoever)
Objects in set is called Elements or Points.
e.g 1. {a, b, c}
2. {1, 2, 3, …. }
3. {x/ x is colour in Rainbow}
4. {<x,y>/x≥0, y ≥ 0} (First Quadrant of Cartesian Plane)
5. (0,1)={x/0<x<1}
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2.  If b is an element of set A then we write 𝑏 ∈ 𝐴
 If b is not an element of set A then we write 𝑏 ∉ 𝐴
 e.g A={Ind, Pak, Ban, SL } and Ind={x/ x is State or Union Territory
of India }
Que. 1. Is Jalgaon ∈ Ind ?
2. Is Maharashtra ∈ A ?
Ans: Both NO
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3. A={1,2,3}, B={3,4,5}
Then 𝐴 ∩ 𝐵 = {3}
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4. A={1,2,3} and B={3,4,5} then B-A= ?
B-A={4, 5}
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5. A={1,2,3}, B={3,4,5}, C={1, 3}
Then C is Subset of A but
C is not Subset of B
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6. •
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7. Function
Definition: “If to each x (in a set S) there corresponds one and only
one value of y, then we say that y is a function of x.“
Only (d) is
Function
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9. Another Definition of function
Defn: Let A and B be any two sets. A function f from (or on) A
into B is a subset of A×B (and hence is a set of ordered pairs
<a,b>) with the property that each a ∈ A belongs to precisely
one pair <a,b>.
 Instead of <x,y> ∈ f we usually write y=f(x).
 Then y is called the image of x under f.
 The set A is called the domain of f.
 The range of f is the set {b ∈ B/ b=f(a) for some a}.
That is, the range of f is the subset of B consisting of all images
of elements of A.
 Such a function is sometimes called a mapping of A into B.
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10. Let f is function from Set A into set B and C ⊂ B then f -1(C) is
defined as {𝑎 ∈ 𝐴/𝑓(𝑎) ∈ 𝐶}
• the set of all points in the domain of f whose images are in C.
• If C has only one point in it, say C= {y}, we usually write f -1(y)
instead of f -1({y}).
• The set f -1(C) is called the inverse image of C under f.
If D ⊂ A, then f(D) is defined as {f(x)/x ∈ D).
• The set f(D) is called the image of D under f.
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11. • Example
Let A={-3,-2,-1,0,1,2,3}, B= {-9,-4,-1,0,1,4,9}
D={-2,0,2} C={-9, 1,4}
Define a function f :A→B by f(x)=x2
OR f={<-3,9>,<-2,4>,<-1,1>,<0,0>,<1,1>,<2,4>,<3,9>}
Then Find
1. Domain 2. Codomain
3. Range 4. f(D)
5. f-1(C) 6. f(-2)
7. f-1(4) 8. f-1(-4)
Answers: 1. A, 2. B, 3. {0,1,4,9}, 4. {0,4}, 5. {1,-1,2,-2}, 6. 4
7. {-2,2}, 8.{} or 𝜑
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12. •
-3
-2
-1
0
1
2
3
-9
-4
-1
0
1
4
9
A B
D
C
f
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14.  Function f(x)= 1+ x3 does not define a function . Why?
Because domain is not explicitly specified.
 Thus the statements
f(x)=1+ x3 (1<x<3) and
g(x)=1+x3 (1<x<4)
define different functions according to our definition.
 In general, suppose f and g are two functions with respective
domains X and Y. If X ⊂ Y and if f(x)=g(x) (x ∈ X),
• we say that g is an extension of f to Y OR
• that f is the restriction of g to X
Y
X
Z
g
f
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16. • If f is a function from A into B, we write f:A → B.
• If the range of f is all of B, we say that f is a function from A onto B.
• In this case we sometimes write f:A⟹ B.
Some Theorems (without proof)
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21. Real Valued Function
• We denote the set of all real numbers by R or ℝ
If f:A→R, we call f a real-valued function.
• If x ∈ A, then f(x) (heretofore called the image of x under f) is also
called the value of f at x.
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22.  For a,b real numbers
• let max(a,b) denote the larger of a and b and
• min(a,b) denote the smaller of a and b.
(If a=b, then max(a,b)=min(a,b)= a=b.)
max(2,3) &
min(e,𝜋) is
3 & e
resp.
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23. Now we Define max(f, g) and min(f, g)
for real-valued functions f, g.
• Defn: If f:A → R, g:A→ R, then max(f, g) is the function defined by max(f,
g)(x)=max[f(x), g(x)] (x ∈ A),
• and min(f, g) is the function defined by
min(f, g) =min[f(x), g(x)] (x ∈ A),
Example : Let f(x) = sin x (0≤ x ≤𝜋/2) and
g(x) = cos x (0≤ x ≤𝜋/2)
Then Find h(x)= max(f, g) and k(x)=min(f, g)
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h(x)= max(f, g)
h(x)= cos x (0≤ x ≤𝜋/4)
= sin x (𝜋/4 ≤ x ≤𝜋/2)
k(x)=min(f, g)
k(x)= sin x (0≤ x ≤𝜋/4)
= cos x (𝜋/4 ≤ x ≤𝜋/2)

25. • Definition : Let x is real Numbers
|x| = x if x > 0
= 0 if x = 0
= -x if x < 0
If f is Real Valued function then |f|(x)=|f(x)|
|f|(x)=|f(x)| = f(x) if f(x) > 0
= f(0) if f(x) = 0
= -f(x) if f(x) < 0
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27. Characteristics Function
• In this section we consider sets which are all subsets of a "big" set S.
• If A ⊂ S, then A’= S - A . For each A ⊂ S we define a function 𝝌𝑨 as
follows.
• Definition: If A ⊂ S, then 𝜒𝐴 (Called the Characteristics function of A) is
defined as
𝜒𝐴(x) = 1 if x ∈ A
𝜒𝐴(x) = 0 if x ∉ A
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28. • The reason for the name "characteristic function”
is obvious--the set A is characterized (completely described) by 𝜒𝐴.
A= B if and only if 𝜒𝐴 = 𝜒𝐵
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0
1
Y
A
𝜒𝐴