Elementary Set Theory

Given a universal set $U$ and two sets $A$ and $B$, demonstrate the application of De Morgan's Laws by simplifying the expression $(A' \cap B')'$, ultimately expressing it in terms of $A$ and $B$ without complements, and specify which fundamental set operation this resultant expression represents between sets $A$ and $B$. The answer is the ______ of A and B.

Within a universal set of complex numbers, let $A$ represent the set of all solutions to the equation $z^3 = 1$, and let $B$ be the set of all solutions to $z^4 = 1$. Determine the cardinality (number of elements) of the set resulting from the intersection, $A \cap B$, and identify the elements comprising this ______.

In the context of set theory, articulate a concise proof demonstrating that for any set $A$, the power set of the empty set, denoted as $P(∅)$, is a subset of the power set of $A$, symbolically represented as $P(∅) \subseteq P(A)$. The proof must explicitly address the composition of $P(∅)$ and its relation to the defining characteristic required for ______.

Given three non-empty sets, $A$, $B$, and $C$, devise a scenario where $A \cup B = A \cup C$ but $B \neq C$. Fully explain the conditions under which this can occur, providing a concrete example using numerically defined sets, thereby illustrating that the union operation does not generally satisfy the cancellation property. The example must show the existence of elements that critically affect the ______.

Formally prove that for any two arbitrary sets $A$ and $B$, the symmetric difference ($A \triangle B$) can be equivalently expressed as $(A \cup B) - (A \cap B)$. The proof must rigorously demonstrate this equivalence using only the fundamental definitions of set union, ______, and set difference.

Given the universal set $U = \mathbb{Z}$ (the set of all integers), and defining two sets as follows: $A = {x \in \mathbb{Z} : x \equiv 1 \pmod{5}}$ and $B = {x \in \mathbb{Z} : x \equiv 3 \pmod{7}}$, determine the cardinality of the complement of the intersection of $A$ and $B$, denoted as $|(A \cap B)'|$, within the subset of $U$ confined to integers from 1 to 100 inclusive. A solution requires calculating highly constrained ______.

Establish a novel theorem positing a condition under which the power set of the union of two sets, $P(A \cup B)$, is equal to the union of their power sets, $P(A) \cup P(B)$. Critically assess and explain the scenario's limiting constraints concerning the universal set and set ______.

Given the universal set $U = {1, 2, 3, ..., 10}$, define set $A$ as the set of prime numbers within $U$, and set $B$ as the set of even numbers within $U$. Construct the set representing the symmetric difference between the complement of $A$ and $B$, expressed as $A' \triangle B$, explicitly listing all elements. This solution requires advanced understanding of number theory, plus set ______.

Consider a real-world scenario managing customer data for a subscription service. Set $A$ represents customers who subscribe to 'Premium Content', Set $B$ represents customers who use 'Mobile Access', and Set $C$ represents customers who have 'Automatic Renewal' enabled. Apply set operations to define and interpret the set representing customers who either subscribe to Premium Content but do not use Mobile Access, or use Mobile Access but do not have Automatic Renewal enabled, or have Automatic Renewal enabled but do not subscribe to Premium Content. Express the resultant set using appropriate notation and simplify wherever possible to test the application of ______ laws.

Given sets $A$ and $B$ within a universal set $U$, use only set identities (Commutative, Associative, Distributive, Identity, Complement, and De Morgan's Laws) to simplify the expression $(A ∪ B) ∩ (A' ∪ B)$ to its simplest equivalent form, and explicitly state which of the aforementioned set identities were crucial in deriving the final ______.