Let A and B be sets. Show that A\B(A\B) ⊆ B.
Not sure how to parse your expression.
B(A\B)
is not correct syntax, since
B is a set
(A\B) is a set, call it C
So, what does BC mean?
Similarly,
(A\B) is a set, so call it D.
Again, what is DC?
two set names juxtaposed has no meaning.
It's actually A \ (A \ B) ⊆ B
To prove that A\B(A\B) ⊆ B, we need to show that every element in the set A\B(A\B) is also in the set B.
Let's break down the given notation A\B(A\B). The symbol '\' represents the set difference operation. So, A\B means the elements that are in set A but not in set B. In this case, A\B(A\B) means taking the set difference of A\B once again.
To show that A\B(A\B) ⊆ B, we can start with an arbitrary element x in A\B(A\B), and then demonstrate that x is also in B.
So, let's assume x is an arbitrary element in A\B(A\B). This means x is in A, but not in B. Now, let's consider the set A\B once again. Since x is in A\B (i.e., x is in A but not in B), it implies that x is not in B.
Now, let's consider the set A\B(A\B). Since x is in A\B(A\B), it indicates that x is in A\B, but not in B. But we already established that x is not in B. Therefore, we can conclude that x is in B, which proves that A\B(A\B) ⊆ B.
In summary, by assuming an arbitrary element x in A\B(A\B) and demonstrating that x is also in B, we have shown that A\B(A\B) ⊆ B.