Prove that (A') '=A by arbitrary elementary method ​

1 Elementary Set Theory

Notation:

{} enclose a set.

{1, 2, 3} = {3, 2, 2, 1, 3} because a set is not defined by order or multiplicity.

{0, 2, 4, . . .} = {x|x is an even natural number} because two ways of writing

a set are equivalent.

∅ is the empty set.

x ∈ A denotes x is an element of A.

N = {0, 1, 2, . . .} are the natural numbers.

Z = {. . . , −2, −1, 0, 1, 2, . . .} are the integers.

Q = {

m

n

|m, n ∈ Z and n 6= 0} are the rational numbers.

R are the real numbers.

Axiom 1.1. Axiom of Extensionality Let A, B be sets. If (∀x)x ∈ A iff x ∈ B

then A = B.

Definition 1.1 (Subset). Let A, B be sets. Then A is a subset of B, written

A ⊆ B iff (∀x) if x ∈ A then x ∈ B.

Theorem 1.1. If A ⊆ B and B ⊆ A then A = B.

Proof. Let x be arbitrary.

Because A ⊆ B if x ∈ A then x ∈ B

Because B ⊆ A if x ∈ B then x ∈ A

Hence, x ∈ A iff x ∈ B, thus A = B.

Definition 1.2 (Union). Let A, B be sets. The Union A ∪ B of A and B is

defined by x ∈ A ∪ B if x ∈ A or x ∈ B.

Theorem 1.2. A ∪ (B ∪ C) = (A ∪ B) ∪ C

Proof. Let x be arbitrary.

x ∈ A ∪ (B ∪ C) iff x ∈ A or x ∈ B ∪ C

iff x ∈ A or (x ∈ B or x ∈ C)

iff x ∈ A or x ∈ B or x ∈ C

iff (x ∈ A or x ∈ B) or x ∈ C

iff x ∈ A ∪ B or x ∈ C

iff x ∈ (A ∪ B) ∪ C

Definition 1.3 (Intersection). Let A, B be sets. The intersection A ∩ B of A

and B is defined by a ∈ A ∩ B iff x ∈ A and x ∈ B

Theorem 1.3. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Proof. Let x be arbitrary. Then x ∈ A ∩ (B ∪ C) iff x ∈ A and x ∈ B ∩ C

iff x ∈ A and (x ∈ B or x ∈ C)

iff (x ∈ A and x ∈ B) or (x ∈ A and x ∈ C)

iff x ∈ A ∩ B or x ∈ A ∩ C

iff x ∈ (A ∩ B) ∪ (A ∩ C)

a and b are the correct answer to what your looking fior

the addition that would strengthen melissa’s thesis statement, both the dark game and the code book explain the importance of codebreaking, is :

main ideas of each book

melissa should discuss the main ideas of each book because these ideas will fully support her thesis statement. through these details, readers will also understand the importance of codebreaking that melissa referred to, which was inspired by the books.

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hope this !

explanation: