Question

Prove the following:

• A'∩(A∪B)=A'∩B

• A'∪(A∩B) = A'∪B

• if A ⊆ B then B ⊆ A

• A−(B−A)=A

Answer #1

Prove the following using Field Axioms of Real
Numbers. prove (b^(−1))^−1=b

Prove or deprove the following sentences:
a . ? – ? = ? ∩ ?
b. (A–B) – C = (A–C) – (B–C)

Prove the following two statements.
a. |b| < a if and only if -a < b < a
b. |a - b| < c if and only if b - c < a < b + c

(7) Prove the following statements.
(c) If A is invertible and similar to B, then B is invertible
and A−1 is similar to B−1 .
(d) The trace of a square matrix is the sum of the diagonal
entries in A and is denoted by tr A. It can be verified that tr(F
G)=tr(GF) for any two n × n matrices F and G. Prove that if A and B
are similar, then tr A = tr B

Prove the following identity on languages A, B, C: A(B ∪
C) = AB ∪ AC
Find a counterexample to the following identity on
languages A, B: A* ∩ B* = (A∩B)*

Are the following languages over {a, b} regular? If they are
then prove it. If they are not prove it with the Pumping Lemma
{an bm | m != n, n >= 0}
{w | w contains the substring ‘aaa’ once and only once }
Clear concise details please, if the language is regular,
provide a DFA/NFA along with the regular expression. Thank you.
Will +1

3. Prove or disprove the following statement: If A and B are
finite sets, then |A ∪ B| = |A| + |B|.

(i)Prove that if
∀a∈A ∃b∈B s.t. a≤b,
then supA≤supB.
(ii)Prove that if
∀a∈A ∃b∈B s.t. a≥b,
then infB≤infA

Prove the following in the plane.
a.) The complement of a closed set is open.
b.) The complement of an open set is closed.

Prove: Let a and b be integers. Prove that integers a and b are
both even or odd if and only if 2/(a-b)

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