I read that the arrow notation $x \rightarrow y$ was invented in the 20th century. Who introduced it? Each map needs both an explicit domain and an explicit codomain (not just a domain, as in previous formulations of set theory, and not just a codomain, as in type theory). -- Lawvere and Rosebrugh Sets for Mathematics, 2003...Read more

I'm a college student just beginning to study the very basic of set theory. In studying about Surjective & Injective functions & how they map their domain to their codomain, it came to my mind a question that I really want to confirm:1) Is it possible to build a "surjective non-injective" function with a domain which has a "lower" cardinality than it's codomain?2) Is it possible to build an "injective non-surjective" function with a domain which has a "higher" cardinality than it's codomain?The context of these questions are: This post ...Read more

Spivak (Calculus, 3e, p. 39) writes: Undoubtedly the most important concept in all of mathematics is that of a function---in almost every branch of modern mathematics functions turn out to be the central objects of investigation.My question is: Would most mathematicians agree with this claim? (I'd like to be able to confidently quote this to high-school students learning about functions and calculus.)...Read more

The question: Let $f: A\rightarrow B$ be a function, and let $\{ B_i : i \in I\}$ be a partition of $B$. Prove that $\{f^{-1}(B_i):i \in I\}$ is a partition of $A$. My work so far: For any given $B_i \in \{ B_i : i \in I\}$, the symbol $f^{-1}(B_i)$ denotes the preimage of $B_i$. The preimage of $B_i$ is the set $\{x \in A: f(x) \in B_i\}$. Since $\{B_i : i \in I\}$ is a partition of $B$, each $b \in B$ belongs to exactly one $B_i$. Also, because $f:A \rightarrow B$ is a function, if $x \in A$ then $f(x) \in B$. Thus if $x \in A$ then $f(x) \in...Read more

Let $A$ be a set such that $|A|=2^{\aleph_0}$ and let $S$ be a partition of $A$ such that $S$ is not countable.Is there a way to define an injective function $f:A\longrightarrow S$ in order to prove that the the cardinality of $S$ is exactly $2^{\aleph_0}$?This is a problem I've encounter solving a much more difficult one, which I actually asked here some days ago but has not been answered yet.This is the last thing I need in order to prove that other problem I mentioned....Read more

Let $f: S \to T$ be a surjective function that maps the elements of each subset $S_{\alpha} \subset S$ to elements $t \in T$. The problem is to show that the collection $\{f^{-1}(\{t\}) = S_{\alpha}\}$ for each $t \in T$ partition the original set $S$.I can show that the $\bigcup S_{\alpha} = S$.How do I show that if $S_{\alpha} \neq S_{\beta}$, then $S_{\alpha} \cap S_{\beta} = \emptyset$?...Read more

I am trying to solve the following problem:Let A and B be sets and let f:A→B be a surjective function. For each b∈ B, let $A_b=(f^{-1}) (\{b\})$. Prove that the collection of sets $\{A_b | b\in B\}$ is a partition of A.I understand that in order for something to be a partition of a set, it must be pairwise disjoint and that $\cup A_b=A$ is also a requirement. I also know that since the function is surjective, A has more elements than B and $f(A)=B$. However, I'm having trouble figuring out how to form this information into a cohesive proof....Read more

Consider the set $(0,1)$ and denote every $a \in (0,1)$ by it's decimal expansion$$a=0.a_1a_2a_3\ldots$$Now, define the equivalence relation $a \sim b$ if and only if $a_p = b_p$ for every prime number $p$; and let $A$ denote the set of equivalence classes.Is it possible to establish an injection $f:[0,1]\rightarrow A$?What I actually want to prove is that is that $|A|=2^{\aleph_0}$. So far, I've been able to prove that $A$ is not countable but that is not enough if one wants to avoid the use of the continuum hypothesis.Is it actually possible ...Read more

Risch, who developed the algorithm in 1968, called it a decision procedure, because it is a method for deciding whether a function has an elementary function as an indefinite integral; and also, if it does, determining it.Is there a way to only decide whether a function has an elementary function as an indefinite integral without determining it?...Read more

I'm trying to obtain the area between the curve of these two functions (for $x>0$), lets call them $f(x)=2\cos(x)$ and $g(x)=x/2$ and my idea is to get the area under the curve of $f(x)$, then subtract the sum of these: the area under the curve of $g(x)$ [$0$, intersection point] and $f(x)$ [intersection point, $\pi/2$]Is this the right way or there's an easier way?Thanks....Read more

I'm trying to find the solution to this because I need to find the area between the curves, but I need this intersection point to properly subtract the unnecessary parts. I know how to do it with polynomials but with 2cosx and x/2 i just don't know what to do.Thanks...Read more

The figure of the problem where $y=\frac x2,y=-\frac x2$ and $x=y^2+1$ enclose the region is this:The shaded region is the enclosed area. Now the max of $x^2+y^2$ in this area is exactly what in this region?Is it the point where $y=\frac{x}{2}$ intersects $x^2+y^2$?...Read more

if a real valued function defined for example as :$[a,b]\times[a,b] \to \mathbb{R}$ , then I ask if there is a function defined as :${[a,b]}^{[a,b]} \to \mathbb{R}$ or Does this ${[a,b]}^{[a,b]}$ make sense in mathematics?Note: $a, b$ are real numbers with $a < b $...Read more

I'm looking for a function that takes an array of values between a..b (like 0..1) and a central point a < c < b (like 0.5) and a factor (like 2, 3, 4, etc) and pushes all the array's values away from the central point.For example if we have [0.0, 0.25, 0.48, 0.56, 0.87, 0.98] as input to the function, I'm expecting to get something like [0.0, 0.23, 0.41, 0.65, 0.895, 0.9848]. The factor number should be able to control how much the numbers are stretched away from the central point.There are some implementation details like how much should...Read more

I have an input graph as below. I am trying to represent the above pictorial representation in mathematical terms. The above pictorial representation can be viewed as a function from X to P (Y), where P (Y) represents the powerset of Y. P (Y) represents the set of all subsets of Y.Is my above mathematical formulation correct for the given picture?...Read more