Write a negation for the following statement:
For all x , if x ∈ A ∩ B then x ∈ B.
To write the negation of the given statement, we need to first determine what the original statement is saying. The original statement is of the form "For all x, if P then Q." This can be translated to "If P is true for any x, then Q is also true for that x."
In the given statement, P is "x ∈ A ∩ B" and Q is "x ∈ B." So, we can interpret the original statement as saying that if an element x is in both set A and set B, then x must also be in set B.
To write the negation, we need to negate the entire statement. The negation of the original statement would be "There exists an x such that x ∈ A ∩ B but x ∉ B." This means that we are claiming there is at least one element x that belongs to both set A and set B, but does not belong to set B.