# Prove $A cup (cap B)=cap{Acup b: b in B}$

Mathematics Asked by Aaron on October 19, 2020

Definition The amalgamated intersection of $$A$$ is defined by $$cap A={x mid forall a in A, x in a }$$

Prove $$A cup (cap B)=cap{Acup b: b in B}$$

Let $$x in cap{A cup b: b in B}$$, then $$forall a, a in Acup b , xin a$$ then $$forall a, a in A$$ $$vee$$ $$a in b$$ then $$forall a, a in A, x in a$$ $$vee$$ $$forall a, a in b, x in a$$ then $$x in A$$ $$vee$$ $$forall a in B, x in a$$ this is beacuase $$a in b wedge b in B Rightarrow a in B$$ and $$x in a wedge a in A Rightarrow x in A$$; then $$x in A cup (cap B)$$

I don’t know if what I did is correct and to prove $$subset$$ I need help. Could you help me? Please.

x in A $$cup$$ $$cap$$B iff
x in A or for all S in B, x in B iff
for all S in B, (x in A or x in S) iff
for all S in B, x in A $$cup$$ S iff
x in $$cap$${ A $$cup$$ S : S in B }

Answered by William Elliot on October 19, 2020

What you write as $$cap{Acup b:bin B}$$, I suspect you mean $${x:forall bin B~.xin Acup b}$$, or simply $$bigcap_{bin B} (Acup b)$$

(Similarly $$bigcap B$$ appears to be a shorthand for $${bigcap}_{bin B} b$$, which is $${x:forall bin B~.xin b}$$ )

Thus you seek to prove: begin{align}Acup bigcap B &= {x: xin Alor xinbigcap B} \[1ex] &={x:xin Alor (forall bin B~.xin b)}\[1ex]&={x:forall bin B~.(xin Alor xin b)}\[1ex]&={x:forall bin B~.xin (Acup b)}\[1ex]&={bigcap}_{bin B}(Acup b)end{align}

Let $$x in cap{A cup b: b in B}$$, then $$forall a, a in Acup b , xin a$$ then $$forall a, a in A$$ $$vee$$ $$a in b$$ then $$forall a, a in A, x in a$$ $$vee$$ $$forall a, a in b, x in a$$ then $$x in A$$ $$vee$$ $$forall a in B, x in a$$ this is beacuase $$a in b wedge b in B Rightarrow a in B$$ and $$x in a wedge a in A Rightarrow x in A$$; then $$x in A cup (cap B)$$

No, $$xin awedge ain A$$ does not imply $$xin A$$.   Moreover you are confusing your $$a$$, $$b$$, and $$x$$ elements.   We just need $$x$$ and $$b$$ — the arbitrary $$x$$ is assumed to be in the thing, and we discuss all $$b$$ that are in $$B$$.

• Take an arbitrary $$x$$ with the assumption that $$xin bigcap_{bin B}(Acap b)$$. That is to say: $$forall bin B~.(xin Acup b)$$. Therefore either $$xin A$$ or if otherwise you can show $$forall bin B~.(xin b)$$.   Hence $$xin Acupbigcap B$$.

Thereby proving : $$bigcap_{bin B}(Acup b)subseteq Acupbigcap B$$.

The converse may be proven through a proof by cases.

• Take an arbitrary $$x$$ with the assumption that $$xin Acupbigcap B$$. That is to say: $$xin A$$ or $$xin bigcap B$$.
• In the case of $$xin A$$, there you can show: $$forall bin B~.xin (Acup b)$$.
• In the case of $$xin bigcap B$$ [that is to say $$forall bin B~.xin b$$], there you can show: $$forall bin B~.xin (Acup b)$$.
• Therefore $$forall bin B~.xin (Acup b)$$ which is to say: $$xinbigcap_{bin B}(Acup b)$$

Thereby proving : $$bigcap_{bin B}(Acup b)supseteq Acupbigcap B$$.

Together proving: $$bigcap_{bin B}(Acup b) = Acupbigcap B$$.

$$blacksquare$$

Answered by Graham Kemp on October 19, 2020

## Related Questions

### Can you find a single solution of this function?

1  Asked on November 20, 2021 by guavas222

### Definition of vertex in graph theory

1  Asked on November 20, 2021

### Gluing Construction of the Grassmanian in Eisenbud/Harris

0  Asked on November 19, 2021 by johnny-apple

### general matrix determinant lemma

1  Asked on November 19, 2021 by silbraz

### Pretty conjecture $x^{left(frac{y}{x}right)^n}+y^{left(frac{x}{y}right)^n}leq 1$

2  Asked on November 19, 2021

### What fragment of ZFC do we need to prove Zorn’s lemma?

3  Asked on November 19, 2021 by zhen-lin

### Let$A$ be a $3times3$ real symmetric matrix such that $A^6=I$ . Then $A^2=I$

2  Asked on November 19, 2021

### Heron’s Formula Intuitive Geometric Proof

1  Asked on November 19, 2021

### For a given circle, prove that the lines of intersections by circles that pass through two given points converge at one point.

1  Asked on November 19, 2021 by taxxi

### Computing the dual change of coordinate matrix $[T^t]^{beta *}_{gamma *}$

2  Asked on November 19, 2021 by ruochan-liu

### A prime ideal is either maximal right ideal or small right ideal.

1  Asked on November 19, 2021 by nirbhay-kumar

### How to evaluate $int_{0}^{infty} x^{nu} frac{e^{-sqrt{x^2+a^2}}}{sqrt{x^2+a^2}} , dx$?

1  Asked on November 19, 2021 by ui-jin-kwon

### Geometric sequences , cones and cylinders

1  Asked on November 19, 2021

### Evaluate $f^{prime prime}(z)$ using Cauchy’s inequality.

2  Asked on November 19, 2021

### Calculation of $left(frac{1}{cos^2x}right)^{frac{1}{2}}$

1  Asked on November 19, 2021 by underdisplayname

### Need help with even number problem

2  Asked on November 19, 2021 by dddb

### Showing an infinite sequence is constant under some condition

1  Asked on November 19, 2021

### Is $mathbb{Q};cong; (prod_{ninomega}mathbb{Z}/p_nmathbb{Z})/simeq_{cal U}$?

3  Asked on November 19, 2021

### Question about dominated convergence: showing $1_{[tau, tau_j)}$ tends to $0$ a.e. for an approximating sequence of stopping times $tau_j$.

1  Asked on November 19, 2021

### Geometric intuition of perpendicular complex vectors

0  Asked on November 19, 2021 by rameesh-paul