Proving (A∪B)^n = (A∪B)∪(A∩B) for Natural N0s n

[solakis1]

Given the definition: \(\displaystyle A^{n+1}=A^n\cup A\) then prove that:

\(\displaystyle (A\cup B)^n =(A\cup B)\cup(A\cap B)\) for all natural N0s n

 

[Evgeny.Makarov]

Given the definition: \(\displaystyle A^{n+1}=A^n\cup A\)

What is $A^1$?

then prove that:

\(\displaystyle (A\cup B)^n =(A\cup B)\cup(A\cap B)\) for all natural N0s n

Hmm, why write $(A\cup B)\cup(A\cap B)$ since $A\cap B$ is included in $A\cup B$?

 

[solakis1]

What is $A^1$?

Hmm, why write $(A\cup B)\cup(A\cap B)$ since $A\cap B$ is included in $A\cup B$?

\(\displaystyle A^1=A\)

Is by definition (or theorem):

1) \(\displaystyle x\in(A\cup B)\Longleftrightarrow x\in A\vee x\in B\)

......Or.......

2)\(\displaystyle x\in(A\cup B)\Longleftrightarrow (x\in A\vee x\in B)\vee (x\in A\wedge x\in B)\)

 

[Evgeny.Makarov]

\(\displaystyle A^1=A\)

Then $A^n=A$ for all $n$.

Is by definition (or theorem):

1) \(\displaystyle x\in(A\cup B)\Longleftrightarrow x\in A\vee x\in B\)

......Or.......

2)\(\displaystyle x\in(A\cup B)\Longleftrightarrow (x\in A\vee x\in B)\vee (x\in A\wedge x\in B)\)

Both.

 

[solakis1]

Then $A^n=A$ for all $n$.

Both.

In that case if we put :

xεA=p, xεB= q,how would we prove :

(pvq)vr =pv(qvr)?

Since pvq <=> (pvq)v(p^q)

 

[Evgeny.Makarov]

In that case if we put :

xεA=p, xεB= q,how would we prove :

(pvq)vr =pv(qvr)?

Associativity of disjunction holds for all propositions, not necessarily x ∈ A and x ∈ B. The way of proving it depends on axiomatics. In Boolean algebra, for example, this is an axiom.

 

[solakis1]

Associativity of disjunction holds for all propositions, not necessarily x ∈ A and x ∈ B. The way of proving it depends on axiomatics. In Boolean algebra, for example, this is an axiom.

By the way:

In the axioms you suggested in Wikipedia ,two of them can be proved using the other axioms.
Those are the axioms of absorption and the axioms of associativity.

I will prove the axioms of absorption and i will leave it to you to prove the axioms of associativity.proof:

1) x^(xvy)

2) (xv0)^(xvy)...using the axiom of identity:av0=a

3) xv(0^y).....using the axiom of distributivity: av(b^c) = (avb)^(avc)

4) xv(y^0)....using the axiom of commutativity: a^b=b^a

5) xv[(y^0)v0]...using the axiom of identity:av0=a

6) xv[(y^0)v(y^y')]...using the axiom of complements: a^a' = 0 (Note a' is the comlement of a)

7) xv[y^(0vy')].....using the axiom of distributivity : a^(bvc) = (a^b)v(a^c)

8) xv[y^(y'v0)].....using the axiom of commutativity: avb=bva

9) xv(y^y') ......using the axiom of identity: av0=a

10) xv0 ......using the axiom of complements :y^y'=0

11) x......using the axiom of identity: av0=aAlso due to the duolity principle we have:

xv(x^y)= x

 

[MarkFL]

...
I will prove the axioms of absorption and i will leave it to you to prove the axioms of associativity...

I'm just curious, why are you leaving work to be done by Evgeny.Makarov?

 

[solakis1]

I'm just curious, why are you leaving work to be done by Evgeny.Makarov?

You mean always or just this time ?

 

[MarkFL]

You mean always or just this time ?

Just this one particular instance, the part of your post that I quoted. I was just curious. :D