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Real Analysis
Let R be reflexive and transitive relation on a set S. Then R intersect R inverse is a
a)reflexive but not transitive relation.
b)transitive but not reflexive relation.
c)symmetric but not reflexive and transitive relation.
d)equivalence relation.
a)reflexive but not transitive relation.
b)transitive but not reflexive relation.
c)symmetric but not reflexive and transitive relation.
d)equivalence relation.
Real Analysis
Solve for the unknown variables:
(a) (a, b) + (3, 1) = (6, 7)
(b) (11, c) + (d, 6) = (-12, −3)
(c) (5, 5) + (e, 4) = (2, f)
(d) (8, −3) − (5, g) = (h, 2)
(a) (a, b) + (3, 1) = (6, 7)
(b) (11, c) + (d, 6) = (-12, −3)
(c) (5, 5) + (e, 4) = (2, f)
(d) (8, −3) − (5, g) = (h, 2)
Real Analysis
Let X and Y be metric spaces and let f:=X→Y be a mapping. Pick out the true statements:
a. if f is uniformly continuous, then the image of every Cauchy sequence in X is a Cauchy sequence in Y ;
b. if X is complete and if f is continuous, then the image of every Cauchy sequence in X is a Cauchy sequence in Y ;
c. if Y is complete and if f is continuous, then the image of every Cauchy sequence in X is a Cauchy sequence in Y
a. if f is uniformly continuous, then the image of every Cauchy sequence in X is a Cauchy sequence in Y ;
b. if X is complete and if f is continuous, then the image of every Cauchy sequence in X is a Cauchy sequence in Y ;
c. if Y is complete and if f is continuous, then the image of every Cauchy sequence in X is a Cauchy sequence in Y
Real Analysis
Pick out the compact sets.
a) {(z1,z2)∈C×C:(z1)2+(z2)2=1}.
b) The unit sphere in ℓ2, the space of all square summable real sequences, with its usual metric d((xi),(yi))=(∑i=1∞|xi−yi|2)1/2.
c) The closure of the unit ball of C1[0,1] in C[0,1].
a) {(z1,z2)∈C×C:(z1)2+(z2)2=1}.
b) The unit sphere in ℓ2, the space of all square summable real sequences, with its usual metric d((xi),(yi))=(∑i=1∞|xi−yi|2)1/2.
c) The closure of the unit ball of C1[0,1] in C[0,1].
Real Analysis
Let (M,d(x,y)) be a metric space and suppose that E, K are nonempty subsets of M with E closed, K compact, and E, K disjoint. Please show that there is a postive real number mu such that d(x,y) >= mu for every x contained in E and y contained in K, using either the definition of compactness in terms of open coverings or the limit point property.
Real Analysis
prove that the sequnce <an> defined by an=3n+7/4n+8 is a monotonic sequence.
Real Analysis
prove that [1/2(a+b)]^2<=1/2(a^2+b^2)
Real Analysis
Give an example of function that is differentiable, its derivative is positive at some point P but there is no neighboorhood of P such that function is monotone there.
Real Analysis
Prove that if x is a rational number and y is an irrational number, then x + y is an irrational
number. If, in addition, x is not equal to 1, then show that xy is an irrational number.
number. If, in addition, x is not equal to 1, then show that xy is an irrational number.
Real Analysis
if a,b,c are real numbers show that min{a,b,c}=min{min{a,b},c}
