Mathematical Logic Problems

Question

1.

Let A and B be sets in a universe U. Prove that the following equality holds:

(A∪B)−(A∩B) = (A−B)∪(B−A)

2.

Let a₁, a₂, ··· be a sequence of real numbers.

We say lim _n→∞ a_n = 11 if for all ∈ > 0, there exist m > 0 such that for all n > m implies |a_n −11| < ∈.

What is the negation of "lim n_→∞ a_n = 11"?



Answer 1

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the x be an element of the set defined by the left side of the equality

then x is in A or B but not both

so if x is in A, then x is not in B. That means x is in A-B

but if x is in B, then x is not in A. That means s is in B-A

so x is either in A-B or B-a which proves the statement --->

<------

Let y be an element of the set defined by the right side of the equality

then x is A but not B or x is in B but not A.

that means x cannot be in both A and B at the same time,

so x is NOT an element of the intersection.

since x is in A or B (but not both) we have

x in the union but not the intersection, which

proves the statement <-----

the sequence does not converge to 11