elementary set theory - Cardinality of Surjective only & Injective only functions

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

Soft Question: Do most mathematicians agree that the function is "the most important concept in all of mathematics"?

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

functions - Problem 4 from Section E of Chapter 12 of Pinter's Book of Abstract Algebra

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

functions - If $A$ is a set of size $2^{\aleph_0}$ and $S$ is an uncountable partition of $A$, is there an injection from $A$ into $S$?

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

elementary set theory - Surjective functions and partitions

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

discrete mathematics - Proving a collection of sets is a partition within a surjective function

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

elementary set theory - Injective function from $(0,1)$ to a partition

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

functions - Area between the curves of $2\cos(x)$ and $x/2$

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

percentages - a Function to Push Numbers Away From a Central Number

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

functions - Is my mathematical formulation correct?

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