Tagged Questions

1
vote
2answers
60 views

Prove $|A| + |B| \ge |A \cup B|$ if $A$ and $B$ are finite sets

I have a question like this: Prove $|A| + |B| \ge |A \cup B|$ if $A$ and $B$ are finite sets. Here is the solution but I don't agree with it: Let $A = \{a_1, \dots, a_n\}$ and $B = \{b_1, \dots, ...
1
vote
1answer
21 views

Counting Card hands with various restrictions

I would like to know if my solutions are correct for the following three combinatorial card questions. In each question, assume we have a standard deck of cards (13 ranks, and 4 suits). How many ...
11
votes
3answers
378 views

prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit

i am given this problem: let $a\ge0$,$b\ge0$, and the sequences $a_n$ and $b_n$ are defined in this way: $a_0:=a$, $b_0:=b$ and $a_{n+1}:= \sqrt{a_nb_n}$ and $b_{n+1}:=\frac{1}{2}(a_n+b_n)$ for all ...
6
votes
4answers
133 views

prove $\lceil{x}\rceil=-\lfloor-x\rfloor$

i am trying to prove that $\lceil{x}\rceil=-\lfloor-x\rfloor$, but having difficulties to prove. the definitions are: $\lceil{x}\rceil:=m-1<x\leq m$ and $\lfloor{x}\rfloor:=n\leq x<n+1$. how ...
6
votes
5answers
184 views

how to prove $\left(\frac{n}{3}\right)^n\leq\frac{1}{3}n!$

i am asked to prove this statement: $$\left(\frac{n}{3}\right)^n\leq\frac{1}{3}n!$$ Now after several attempts, i am lost not knowing where and how to start. if I use induction, i am stuck on ...
2
votes
1answer
133 views

prove with $\epsilon$-$\delta$-argument: $x\rightarrow |-2x+3|$ is continuous

i am asked to prove with $\epsilon$-$\delta$-argument that $x\rightarrow |-2x+3|$ is continuous my steps: Definition of $\epsilon-\delta$-argument: $\forall \epsilon >0 \exists \delta>0$ with ...
4
votes
4answers
192 views

Proof of $n^2 \leq 2^n$.

I am trying to prove that $n^2 \leq 2^n$ for all natural $n$ with $n \ne 3$. My steps are: induction base case: $n=0:$ $0² \leq 2⁰$ which is okay. inductive step: $n \rightarrow n+1:$ ...
7
votes
2answers
98 views

Show $\lim_{n \to \infty} \min\{a_{n},b_{n}\} = \min\{a,b\}$

If $\lim_{n \to \infty} a_{n} = a$ and $\lim_{n \to \infty} b_{n} = b$, how can we show that $\lim_{n \to \infty} \min\{a_{n},b_{n}\} = \min\{a,b\}$? I say $\min\{a_{n},b_{n}\} $ has two cases: ...
9
votes
5answers
462 views

Prove $|a+b|+|a-b| \geq |a|+|b|$

I am fighting with this proof-writing problem for a while. The statement says $$|a+b|+|a-b| \geq |a|+|b|.$$ I know the triangle inequality which says$$|a+b| \leq |a|+|b|.$$ How can I use this ...
1
vote
2answers
77 views

how to prove this simple statement

i am trying to prove this statement. for any $a,b \in \mathbb{R}$, $$\max\{a,b\}=\frac{1}{2}\big(a+b+|a-b|\big)$$ and $$\min\{a,b\}=\frac{1}{2}\big(a+b-|a-b|\big)$$ i am eating myself not knowing ...
1
vote
1answer
66 views

simple statement proof

i am proving this statement about strict isotoneness. i will try on my own and you will tell me whether i am okay or not :) $A$ is subset of $\mathbb{R}$ $f$ is strict isotone $ \Longleftrightarrow ...
2
votes
2answers
64 views

complex numbers - proof of this statement

i am trying to prove this statement, i dont but how to start. $$\forall z,w \in \mathbb{C}\quad |z|^2+|w|^2=\frac{1}{2}(|z+w|^2+|z-w|^2)$$ can someone please show me how start?
0
votes
1answer
38 views

what is the difference - sorry for over-simplicity

i am asking too simple question, sorry for that. what is the difference between these two imaginär numbers? $\operatorname{Im}(| \sqrt2+3i|^2)$ vs. $\operatorname{Im}((\sqrt2+3i)^2)$ $| ...
1
vote
3answers
54 views

minimum of this simple set

i need again some help here. i am defining the minimum and max and inf and sup of this set $A:=(]1,2[ \cup ]2,3]) \cup \{2\}$ which is equal to the interval $(1,3]$ i say, max is 3, and sup is also ...
2
votes
3answers
236 views

archimedean Property - proof

i am stumbling across this statement. i need to show minimum, maximum, infimum and supremum, if they exist. $$ C:= \bigcup_{n \in \mathbb{N}} [0,1/n[$$ the archimedean property says: let $e,x$ be ...
3
votes
2answers
356 views

Proof by induction for Fibonacci numbers

How can we prove by induction the following? $ F_{n+1} = \left\{ \begin{array}{l l} F_{n/2}^2+F_{(n+2)/2}^2 & \quad \text{if $n$ is even}\\ ...
1
vote
1answer
433 views

fibonacci numbers - induction proof

i am trying to prove this statement of fibonacci numbers by induction, i am stuck though on the way: my steps: definition: $F_{0}:=0, F_{1}:=1 $ and $F_{n}:=F_{n-1}+F_{n-2}$ The Hypothesis is: ...
0
votes
2answers
58 views

Maps - question about $f(A \cup B)=f(A) \cup f(B)$ and $ f(A \cap B)=f(A) \cap f(B)$

I am struggling to prove this map statement on sets. The statement is: Let $f:X \rightarrow Y$ be a map. i) $\forall_{A,B \subset X}: f(A \cup B)=f(A) \cup f(B)$ ii) $\forall_{A,B \subset X}: ...
0
votes
1answer
26 views

All solutions for a homogene equations

Suppose we have an equation of the form $ax+by=0$ with $a,b,c \in \mathbb{Z}$. For simplicity, $a \neq 0, b \neq 0$. Then, a single solution to this equation is $(x_0, y_0)=(-a, b)$. My book states ...
2
votes
1answer
85 views

Is this relation transitive if $n=m$?

If $X$ is a set and $n \in \mathbb N$, then $[X]^n$ will denote the set of all subsets of $X$ with exactly $n$ elements. For a set $X$ and natural numbers $n$ and $m$ define a relation $R$ on $[X]^n$ ...