Homework Statement
Prove that set of all onto mappings of A->A is closed under composition of mappings:
Homework Equations
Definition of onto and closure on sets.
The Attempt at a Solution
Say, ##f## and ##g## are onto mappings from A to A.
Now, say I have a set S(A) = {all onto mappings of A...
Homework Statement
I need to show that $$\frac{x}{x^2+1}$$ is either onto or not.
My domain is $$R-{0}$$ and range is $$R$$
Homework Equations
I have learn to do this to show that a function is surjective
y = $$\frac{x}{x^2+1}$$ and solve for x, but I am not sure how to proceed here.
The...
Homework Statement
Imagine that force for is atom was ## F= - \frac{\beta}{r^4}##, rather than ##F=- \frac{ke^2}{r^2}##, and consider only circular orbits, it would remain true that ##L_n= n \hbar##
a.) From Netwon's law find the relationship between ##T ##(Kinetic Energy) and ##V##,
b.) Find...
Homework Statement
Title of pie chart: New Construction in Daisy Hill Subdivision
Given a pie chart with sections:
Currently Completed: 26%
Currently Under Construction: 42%
Approved, but Not Yet Started: 32%
Question:
When construction is completed in Daisy Hill Subdivision there'll...
Homework Statement
Let's assume that the classical ideas of space and time are correct, so that there could only be one frame, "ether", in which light traveled with same speed in all directions.
Assume that the earth's speed relative to the ether frame is our orbital speed around the sun...
Homework Statement
At time t=0, a block is released from point O on the slope shown in the figure.
The block accelerates down the slope, overcoming sliding friction.
a.) Choose axes 0xy as shown, and solve the equation ##\Sigma F = m a## into its x and y components.
Hence find the block's...
Hello all,
I have a bit of a dilemma.
I have two free electives to take and I am considering taking two "advanced" math classes and get a math minor or two semi-good classes from whatever.
If I take two math classes and I get A's, it would be lovely. But, If I get low grades, my GPA will...
Homework Statement
Prove that
## lim_{x\implies 1} \frac{2}{x-3} = -1 ##
Use delta-epsilon.
The Attempt at a Solution
Proof strategy:
## | { \frac{ 2}{x-3} +1 } | < \epsilon ##
## \frac{x-1}{x-3} < \epsilon ##
, since delta have to be a function of epsilon alone and not include x. I...
Homework Statement
a.) Prove: If an integer ##a## does not divide ##bc##, then ##a## does not divide ##b## and ##a## does not divide ##c##.
b.) State and either prove or disprove the converse of the above statement.
The Attempt at a Solution
a.) Proof by contrapositive
## a|c \vee a|b...
Homework Statement
Prove for any ##a,b \in \mathbb{R^+} : |a-b| \leq \sqrt{a^2 +b^2} ##
The Attempt at a Solution
How should I start? Can I use induction?
Should I use contrapositive?
Homework Statement
Let ##n \in \mathbb{Z} , n \not | 3##. Prove that ##gcd( n , n +3 ) =1 ##
The Attempt at a Solution
If n is not divisible by 3, then
n = 3k+1 or n =3k+2 , ## k \in \mathbb{Z} ##
What is a feasible approach? Can I do this?
For first case,
## gcd(3k+1, 3k+4 ) = 1 \\...
Homework Statement
Let ## S = \{ (m,n) : m,n \in \mathbb{N} \} \\ ##
a.) Show function ## f: S -> \mathbb{N} ## defined by ## f(m,n) = 2^m 3^n ## is injective
b.) Use part a.) to show cardinality of S.
The Attempt at a Solution
a.) ## f(a,b) = f(c, d ) ; a,b,c,d \in \mathbb{N} \\\\ 2^a...
Homework Statement
a.) Prove ## \exists x, x \in \mathbb{R} | x^3 -x^2 = 5##
I know that x = 2.1163, but how do I find this without a calculator?
b.) Prove that ## \not \exists x, x \in \mathbb{R} | x^4 - 2x^2 +2 =0##
The Attempt at a Solution
x^2 ( x-1 )=5 for part a)...
Homework Statement
Prove that the open interval (1,3) and the closed interval [1,4] have the same cardinality.
Homework Equations
The Attempt at a Solution
I have to prove bijection.
The injective part is obvious.
Say, A =(1,3) and B =[1,4]
f: A → B
f(x) = x
It's...
Homework Statement
Give an example of a set S of four (distinct) positive integers such that the greatest common divisor of all
six pairs of elements of S is 6.
Homework Equations
The Attempt at a Solution
Can I say that my numbers are in the form?
6
12
18
30
Is this ok?
Set Theory -- Uncountable Sets
Homework Statement
Prove or disprove.
There is no set A such that ##2^A## is denumberable.
The Attempt at a Solution
A set is denumerable if ##|A| = |N|##
My book shows that the statement is true.
If A is denumerable, then since ##|2^A| > |A|, 2^A ##...
Homework Statement
Explain why ## (0,1 ) ## and ## (0,2)##
I have proved that a function ## f: (0,1 ) -> (0,2) ##defined by f(x) =2x is bijective.
Homework Equations
The Attempt at a Solution
I could state that due to the sets relationship being bijective they have the same cardinality...
Homework Statement
Let ## C= \{ x \in R : x \geq 1 \} ## and ## D = R^+ ##
For each f defined below, determine ## f(C), f^{-1}(C), f^{-1}(D), f^{-1} (\{1\}) ##
a.) ## f: R -> R ## is defined by ## f(x) =x^2##
I have problems with the definitions
The Attempt at a Solution
a.)
## f(C)= { 1 ...
Homework Statement
a.) Is symmetric and transitive, but not reflective:
b.) consists of exactly 8 ordered pairs and is symmetric and transitive:
The Attempt at a Solution
If the question asks me to define some relation, do I need to define some math property like power of some number or...
Homework Statement
Let ## H = \{ 2^{m} : m \in Z\}##
A relation R defined in ##Q^{+} ## by ##aRb ##, if ## \frac{a}{b} \in H##
a.) Show that R is an equivalence Relation
b.) Describe the elements in the equivalence class [3].
The Attempt at a Solution
For part a, I think I am able to solve...
Homework Statement
Prove Existence Unique Real Solution to
## x^{3} + x^{2} -1 =0 ## between ## x= \frac{2}{3} \text{and} x=1##
The Attempt at a Solution
## x^{2} ( x+1) =1 ##
I know that the solution is x =0.75488, but this came from some website. How do I find this number without a...
Homework Statement
Let n be any odd integer. Prove that 1 is the only "common" divisor of the integers n and n+2.
The Attempt at a Solution
I don't think I understand the question.
The few notes I have state d| (n+2 )- n
This resembles n+2 ##\equiv## n mod d , but I don't see the...
Homework Statement
Suppose n is an integer which is not a divisor of 5.
Prove that ##n^{4} \equiv 1 mod5##
The Attempt at a Solution
I know that 16 mod 5 is equivalent to 1 mod5.
##16 = 2^{4}##
2 is not a divisor of 5. How do I prove this for the general case?
I know
##n^{4} -1...
Homework Statement
Prove that for every two real numbers x and y
##|x+y| \leq |x| + |y| ##
Homework Equations
The Attempt at a Solution
There are three cases. The easiest ones is when they are both positive and negative.
The third one I have problems with.
The numbers have different...
Homework Statement
Let ##n\in N ##. Prove that if ##|n-1|+|n+1|\leq 1,## then ## |n^{2}-1|\leq 4##
Homework Equations
The Attempt at a Solution
I am trying to show by a counter example that this statement is not true.
Consider this statement:
##|n-1|+|n+1| \leq 1##
Assume ...
Homework Statement
In each of the two following open sentences P(x) and Q(x) over a domain S are given.
Determine all ##x \in S## for which P(x) → Q(x) is a true statement.
## P(x): x \in [-1, 2]; Q(x): x^{2} \leq 2; S=[-1,1] ##
Homework Equations
According to truth values for →:
a...
Homework Statement
I have found the roots of my polynomial:
## (2x+3y)^{2}-1 =0 ##
Roots are x=3n+2 & y=-2n-1, where n belongs to all Z.
What does it mean that the solution has arbitrary large coordinates?
The Attempt at a Solution
I think I know the basic concept of root. It could be...