For Exercises 9–20, refer to the sets A , B , C , and D . Determine the union or intersection as indicated. Express the answer in interval notation, if possible. (See Example 2 .) A = { x | x < − 4 } , B = { x | x > 2 } , C = { x | x ≥ − 7 } , D = { x | 0 ≤ x < 5 } A ∩ D
For Exercises 9–20, refer to the sets A , B , C , and D . Determine the union or intersection as indicated. Express the answer in interval notation, if possible. (See Example 2 .) A = { x | x < − 4 } , B = { x | x > 2 } , C = { x | x ≥ − 7 } , D = { x | 0 ≤ x < 5 } A ∩ D
Solution Summary: The author explains the formula used to find the union or intersection of two given sets.
For Exercises 9–20, refer to the sets A, B, C, and D. Determine the union or intersection as indicated. Express the answer in interval notation, if possible. (See Example 2.)
A
=
{
x
|
x
<
−
4
}
,
B
=
{
x
|
x
>
2
}
,
C
=
{
x
|
x
≥
−
7
}
,
D
=
{
x
|
0
≤
x
<
5
}
ے ملزمة احمد
Q (a) Let f be a linear map from a space X into a space Y and (X1,X2,...,xn) basis for X, show that fis one-to-
one iff (f(x1),f(x2),...,f(x) } linearly independent.
(b) Let X= {ao+ax₁+a2x2+...+anxn, a;ER} be a vector space over R, write with prove a hyperspace and a
hyperplane of X.
مبر خد احمد
Q₂ (a) Let M be a subspace of a vector space X, and A= {fex/ f(x)=0, x E M ), show that whether A is
convex set or not, affine set or not.
Write with prove an
application of Hahn-Banach theorem.
Show that every singleton set in a normed space X is closed and any finite set in X is closed (14M)
Let M be a proper subspace of a finite dimension vector space X over a field F show that
whether: (1) If S is a base for M then S base for X or not, (2) If T base for X then base for M
or not.
(b) Let X-P₂(x) be a vector space over polynomials a field of real numbers R, write with L
prove convex subset of X and hyperspace of X.
Q₂/ (a) Let X-R³ be a vector space over a over a field of real numbers R and
A=((a,b,o), a,bE R), A is a subspace of X, let g be a function from A into R such that
gla,b,o)-a, gEA, find fe X such that g(t)=f(t), tEA.
(b) Let M be a non-empty subset of a space X, show that M is a hyperplane of X iff there
Xiff there
exists fE X/10) and tE F such that M=(xE X/ f(x)=t).
(c) Show that the relation equivalent is an equivalence relation on set of norms on a space
X.
Q/(a)Let X be a finite dimension vector space over a field F and S₁,S2CX such that S₁SS2. Show that
whether (1) if S, is a base for X then base for X or not (2) if S2 is a base for X then S, is a base for X or not
(b) Show that every subspace of vector space is convex and affine set but the conevrse need not to be true.
allet M be a non-empty subset of a vector space X over a field F and x,EX. Show that M is a
hyperspace iff xo+ M is a hyperplane and xo€ xo+M.
bState Hahn-Banach theorem and write with prove an application about it.
Show that every singleten subset and finite subset of a normed space is closed.
Oxfallet f he a function from a normad roace YI
Show tha ir continuour aty.GYiff
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RELATIONS-DOMAIN, RANGE AND CO-DOMAIN (RELATIONS AND FUNCTIONS CBSE/ ISC MATHS); Author: Neha Agrawal Mathematically Inclined;https://www.youtube.com/watch?v=u4IQh46VoU4;License: Standard YouTube License, CC-BY