Problems to demonstrate a succesion of sets
$begingroup$
The exercise seems easy but i have problems to resolve it:
We define a succession of sets:
$$begin{align}
A_k &= {{m in mathbb N | m < n} | n in mathbb N, n le k} (text{for each } k in mathbb N)\
B &= {{m in mathbb N | m < n} | n in mathbb N}
end{align}$$
Demonstrate that $A_k subseteq B$ for all $k in mathbb N$.
I tried to do it using simple induction but for base case $k = 0$ I dont know how to demonstrate that $A_0 subseteq B$.
induction
edited Dec 2 '18 at 10:38
mrtaurho
5,46041237
asked Dec 2 '18 at 10:32
jackes gamerojackes gamero
1
$endgroup$
add a comment |
$begingroup$
The exercise seems easy but i have problems to resolve it:
We define a succession of sets:
$$begin{align}
A_k &= {{m in mathbb N | m < n} | n in mathbb N, n le k} (text{for each } k in mathbb N)\
B &= {{m in mathbb N | m < n} | n in mathbb N}
end{align}$$
Demonstrate that $A_k subseteq B$ for all $k in mathbb N$.
I tried to do it using simple induction but for base case $k = 0$ I dont know how to demonstrate that $A_0 subseteq B$.
induction
edited Dec 2 '18 at 10:38
mrtaurho
5,46041237
asked Dec 2 '18 at 10:32
jackes gamerojackes gamero
1
$endgroup$
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Dec 2 '18 at 10:36
$begingroup$
For $k=0$ the succession $A_0$ has only one member, because there is only one $n le 0$, i.e. $0$ itself. Thus $A_0 = { { m in mathbb N mid m < 0 } } = { emptyset }$.
$endgroup$
– Mauro ALLEGRANZA
Dec 2 '18 at 10:40
$begingroup$
To show $A_0 subseteq B$, you just need to show that every element of $A_0$ is also an element of $B$. Can you list the elements of $A_0$ and show that they are all in $B$?
$endgroup$
– platty
Dec 2 '18 at 10:41
$begingroup$
Thats actually my problem. if A0 = {∅} Does {∅} ⊆ B?
$endgroup$
– jackes gamero
Dec 2 '18 at 11:04
add a comment |
$begingroup$
The exercise seems easy but i have problems to resolve it:
We define a succession of sets:
$$begin{align}
A_k &= {{m in mathbb N | m < n} | n in mathbb N, n le k} (text{for each } k in mathbb N)\
B &= {{m in mathbb N | m < n} | n in mathbb N}
end{align}$$
Demonstrate that $A_k subseteq B$ for all $k in mathbb N$.
I tried to do it using simple induction but for base case $k = 0$ I dont know how to demonstrate that $A_0 subseteq B$.
induction
edited Dec 2 '18 at 10:38
mrtaurho
5,46041237
asked Dec 2 '18 at 10:32
jackes gamerojackes gamero
1
$endgroup$
The exercise seems easy but i have problems to resolve it:
We define a succession of sets:
$$begin{align}
A_k &= {{m in mathbb N | m < n} | n in mathbb N, n le k} (text{for each } k in mathbb N)\
B &= {{m in mathbb N | m < n} | n in mathbb N}
end{align}$$
Demonstrate that $A_k subseteq B$ for all $k in mathbb N$.
I tried to do it using simple induction but for base case $k = 0$ I dont know how to demonstrate that $A_0 subseteq B$.
induction
induction
edited Dec 2 '18 at 10:38
mrtaurho
5,46041237
asked Dec 2 '18 at 10:32
jackes gamerojackes gamero
1
edited Dec 2 '18 at 10:38
mrtaurho
5,46041237
asked Dec 2 '18 at 10:32
jackes gamerojackes gamero
1
edited Dec 2 '18 at 10:38
mrtaurho
5,46041237
edited Dec 2 '18 at 10:38
mrtaurho
5,46041237
edited Dec 2 '18 at 10:38
mrtaurho
5,46041237
5,46041237
asked Dec 2 '18 at 10:32
jackes gamerojackes gamero
1
asked Dec 2 '18 at 10:32
jackes gamerojackes gamero
1
asked Dec 2 '18 at 10:32
jackes gamerojackes gamero
1
1
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Dec 2 '18 at 10:36
$begingroup$
For $k=0$ the succession $A_0$ has only one member, because there is only one $n le 0$, i.e. $0$ itself. Thus $A_0 = { { m in mathbb N mid m < 0 } } = { emptyset }$.
$endgroup$
– Mauro ALLEGRANZA
Dec 2 '18 at 10:40
$begingroup$
To show $A_0 subseteq B$, you just need to show that every element of $A_0$ is also an element of $B$. Can you list the elements of $A_0$ and show that they are all in $B$?
$endgroup$
– platty
Dec 2 '18 at 10:41
$begingroup$
Thats actually my problem. if A0 = {∅} Does {∅} ⊆ B?
$endgroup$
– jackes gamero
Dec 2 '18 at 11:04
add a comment |
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Dec 2 '18 at 10:36
$begingroup$
For $k=0$ the succession $A_0$ has only one member, because there is only one $n le 0$, i.e. $0$ itself. Thus $A_0 = { { m in mathbb N mid m < 0 } } = { emptyset }$.
$endgroup$
– Mauro ALLEGRANZA
Dec 2 '18 at 10:40
$begingroup$
To show $A_0 subseteq B$, you just need to show that every element of $A_0$ is also an element of $B$. Can you list the elements of $A_0$ and show that they are all in $B$?
$endgroup$
– platty
Dec 2 '18 at 10:41
$begingroup$
Thats actually my problem. if A0 = {∅} Does {∅} ⊆ B?
$endgroup$
– jackes gamero
Dec 2 '18 at 11:04
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Dec 2 '18 at 10:36
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Dec 2 '18 at 10:36
$begingroup$
For $k=0$ the succession $A_0$ has only one member, because there is only one $n le 0$, i.e. $0$ itself. Thus $A_0 = { { m in mathbb N mid m < 0 } } = { emptyset }$.
$endgroup$
– Mauro ALLEGRANZA
Dec 2 '18 at 10:40
$begingroup$
For $k=0$ the succession $A_0$ has only one member, because there is only one $n le 0$, i.e. $0$ itself. Thus $A_0 = { { m in mathbb N mid m < 0 } } = { emptyset }$.
$endgroup$
– Mauro ALLEGRANZA
Dec 2 '18 at 10:40
$begingroup$
To show $A_0 subseteq B$, you just need to show that every element of $A_0$ is also an element of $B$. Can you list the elements of $A_0$ and show that they are all in $B$?
$endgroup$
– platty
Dec 2 '18 at 10:41
$begingroup$
To show $A_0 subseteq B$, you just need to show that every element of $A_0$ is also an element of $B$. Can you list the elements of $A_0$ and show that they are all in $B$?
$endgroup$
– platty
Dec 2 '18 at 10:41
$begingroup$
Thats actually my problem. if A0 = {∅} Does {∅} ⊆ B?
$endgroup$
– jackes gamero
Dec 2 '18 at 11:04
$begingroup$
Thats actually my problem. if A0 = {∅} Does {∅} ⊆ B?
$endgroup$
– jackes gamero
Dec 2 '18 at 11:04
add a comment |
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$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Dec 2 '18 at 10:36
$begingroup$
For $k=0$ the succession $A_0$ has only one member, because there is only one $n le 0$, i.e. $0$ itself. Thus $A_0 = { { m in mathbb N mid m < 0 } } = { emptyset }$.
$endgroup$
– Mauro ALLEGRANZA
Dec 2 '18 at 10:40
$begingroup$
To show $A_0 subseteq B$, you just need to show that every element of $A_0$ is also an element of $B$. Can you list the elements of $A_0$ and show that they are all in $B$?
$endgroup$
– platty
Dec 2 '18 at 10:41
$begingroup$
Thats actually my problem. if A0 = {∅} Does {∅} ⊆ B?
$endgroup$
– jackes gamero
Dec 2 '18 at 11:04