A combinatorial question: Let $S = \{1,2,3,4\}$ and $X = \{f:S\rightarrow S | x<y \implies f(x)\leq f(y)\}.$ Find $|X|$.
113329
Let $S=\{1,2,3,4,\dotsc,N\}$ and $X=\{ f: S \rightarrow S \mid x < y \Rightarrow f(x)\leqslant f(y)\}$. Then $|X|$ is equal to what?
23125
Prove that there is a bijection between the set of all subsets of $X$, $P(X)$, and the set of functions from $X$ to $\{0,1\}$.
52095
Prove either $G=ST$ or |$G|\geq|S|+|T|$
122944
Prove $|y''(x)| \leq 40$ for all $x \in [1,3]$.
35889
Suppose that f(0) = f(1) = 0 and x ∈ (0, 1). Prove that there exists c ∈ (0, 1) such that f'(c) = f(x).
125788
A function with a property $f(x+y)=f(x)f(y)$
98921
Does $y(y+1) \leq (x+1)^2$ imply $y(y-1) \leq x^2$?
88278
Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective
14975
Functions satisfying $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$
31732
If the function $f$ satisfies the equation $f(xf(y)+x)=xy+f(x)$, find $f$
72694
Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
133504
Suppose $X$ is a metric space and $S \subseteq X.$ Then, $S^o=\{x \in X~|~dist(x,S^c)>0\}$.
90565
If $f:[0,1] \rightarrow \mathbb{R}$ is continuous function such that $f(0)=f(1)$ then there exists $ x\in [0,1]$ such that $f(x) = f(x+\frac{1}{n})$.
135694
Proving $\kappa^{\lambda} = |\{X: X \subseteq \kappa, |X|=\lambda\}|$
83996
For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$
48765
Let $S$ and $T$ be finite non-empty sets such that $|S| = |T|$. Show that the function $f : S\to T$ is onto if and only if it is one-to-one.
144222
If $f(x + 1) + f(x − 1) = f(x), \forall x \in \mathbb{R}$,then how to find $k$ such that $f(x + k) = f(x)$?