# Thread: Problem 1, Section 1...:(

1. ## Problem 1, Section 1...:(

Explain why $\displaystyle 2\in\{1,2,3\}$.

2. Originally Posted by VonNemo19
Explain why $\displaystyle 2\in\{1,2,3\}$.
Is this a trick question?
What don't understand about it? Do you know the symbols involved?

3. Originally Posted by Plato
Is this a trick question?
What don't understand about it? Do you know the symbols involved?

In particular:

"According to cantor, a set is made up of objects called members or elements...If the first blank in '_____ is a member of _____' is filled in with the name of an object, and the second with the name of a set, the resulting sentence can be classified as either true or false."

So, I guess the answer is

Because $\displaystyle 2\in\{1,2,3\}$ is a true statement.

Right?

4. Explain why
Because is a true statement.
The questions "Why ?" and "Why '' is a true statement" are, in fact, one and the same. If somebody asks you, "Why are you late?", it does not make sense to answer "Because 'I am late' is a true statement".

5. Originally Posted by emakarov
The questions "Why ?" and "Why '' is a true statement" are, in fact, one and the same. If somebody asks you, "Why are you late?", it does not make sense to answer "Because 'I am late' is a true statement".
So, what would you say they are looking for here?

6. I am also not sure about exact words for such a trivial case, but I would say, because 2 is listed as an element of the set. That's why it belongs.

Unless it's advanced math, the concepts of a set, an element and belonging are taken as primitive.

7. Originally Posted by VonNemo19
Explain why $\displaystyle 2\in\{1,2,3\}$.
Here is the proof:

$\displaystyle 2=2\Longrightarrow 2=2\vee 2=1\vee 2=3\Longrightarrow 2=1\vee\ 2=2\vee 2=3\Longrightarrow 2\in\{1,2,3\}$.

Since $\displaystyle x\in\{a,b,c\}\Longleftrightarrow x= a\vee x= b\vee x= c$

8. Hello at all!

Here is a formal proof for the statement:

1) $\displaystyle \forall x(x=x)$.................................................. .................................................. .law of identity
2) $\displaystyle 2=2$.................................................. .................................................. ............1, Universal Elimination, where we put $\displaystyle x=2$
3) $\displaystyle 2=2\vee 2=1$.................................................. .................................................. ....2, Disjunction Introduction
4) $\displaystyle 2=1\vee 2=2$.................................................. .................................................. ....3, Commutation equivalence
5) $\displaystyle \forall x\forall y\forall z[z\in \{x,y\}\Longleftrightarrow z=x\vee z=y]$.................................................. ...........theorem of set theory
6) $\displaystyle 2\in \{1,2\}\Longleftrightarrow 2=1\vee 2=2$.................................................. ......................5, Universal Elimination, where we put $\displaystyle x=1$, $\displaystyle y=2$ and $\displaystyle z=2$
7) $\displaystyle 2=1\vee 2=2\Longrightarrow 2\in \{1,2\}$.................................................. ..........................6, Biconditional Elimination
8) $\displaystyle 2\in \{1,2\}$.................................................. .................................................. ...4, 7, Modus Ponens
9) $\displaystyle 2\in \{1,2\}\vee 2\in \{3\}$.................................................. ......................................8, Disjunction Introduction
10) $\displaystyle \forall x\forall A\forall B[x\in A\cup B\Longleftrightarrow x\in A\vee x\in B]$.................................................. ....theorem of set theory
11) $\displaystyle 2\in \{1,2\}\cup \{3\}\Longleftrightarrow 2\in \{1,2\}\vee 2\in \{3\}$.............................................10, Universal Elimination, where we put $\displaystyle x=2$, $\displaystyle A=\{1,2\}$ and $\displaystyle B=\{3\}$
12) $\displaystyle 2\in \{1,2\}\vee 2\in \{3\}\Longrightarrow 2\in \{1,2\}\cup \{3\}$.................................................1 1, Biconditional Elimination
13) $\displaystyle 2\in \{1,2\}\cup \{3\}$.................................................. ..........................................9, 12, Modus Ponens
14) $\displaystyle \forall x\forall y\forall z[\{x,y,z\}=\{x,y\}\cup \{z\}]$.................................................. ........definition of triplet set
15) $\displaystyle \{1,2,3\}=\{1,2\}\cup \{3\}$.................................................. .....................................14, Universal Elimination, where we put $\displaystyle x=1$, $\displaystyle y=2$ and $\displaystyle z=3$
16) $\displaystyle \forall X\forall Y[X=Y\Longleftrightarrow \forall x(x\in X\Longleftrightarrow x\in Y)]$............................axiom of extensionality
17) $\displaystyle \{1,2,3\}=\{1,2\}\cup \{3\}\Longleftrightarrow \forall x[x\in \{1,2,3\}\Longleftrightarrow x\in \{1,2\}\cup \{3\}]$..............16, Universal Elimination, where we put $\displaystyle X=\{1,2,3\}$ and $\displaystyle Y=\{1,2\}\cup \{3\}$
18) $\displaystyle \{1,2,3\}=\{1,2\}\cup \{3\}\Longrightarrow \forall x[x\in \{1,2,3\}\Longleftrightarrow x\in \{1,2\}\cup \{3\}]$..............17, Biconditional Elimination
19) $\displaystyle \forall x[x\in \{1,2,3\}\Longleftrightarrow x\in \{1,2\}\cup \{3\}]$................................................15 , 18, Modus Ponens
20) $\displaystyle 2\in \{1,2,3\}\Longleftrightarrow 2\in \{1,2\}\cup \{3\}$.................................................. ........19, Universal Elimination, where we put $\displaystyle x=2$
21) $\displaystyle 2\in \{1,2\}\cup \{3\}\Longrightarrow 2\in \{1,2,3\}$.................................................. ............20, Biconditional Elimination
22) $\displaystyle 2\in \{1,2,3\}$.................................................. .................................................. 13, 21, Modus Ponens

Best wishes,
Seppel