Question

Linear Algebra:

Answer 1

Explain, if

B = A intersects B, then \[B \subseteq A\] and conversely

Answer 2

-->
\(B= B\cap A\)
1) \(B\subseteq (B\cap A) \)
Let x \(\in B\) , x \(\in B\cap A\) = x \(\in B \) and x \(\in A\)
which shows \(B\subseteq A\)

Answer 3

Visualize it by Venn.

Answer 4

How did you show, B is a sub set of (B intersection A) ?

Answer 5

I understand that for X contained in B, X contained in B intersect A, then X is contained in A

Answer 6

\[B= B\cap A\]
Means \[B\subseteq (B\cap A) ~~and~~ (B\cap A)\subseteq B\]

Answer 7

That's why on top I put --> , it means I prove from left to right,
next you can use from right to left <--- , to prove "conversely"

Answer 8

Conversely I would say,
\[x \in B = BintersectionA\]?

Answer 9

conversely on this problem means
if \(B\subseteq A\) then B = \(B \cap A\)

Answer 10

Right so first I would say X in B and since B is a subset of A
X in A and which means B = B intersect A ?

Answer 11

for the conversely

Answer 12

yes, something like this, just add some more lines.