1. The problem involves understanding set theory notation and relationships between sets such as $\mathbb{Q}$ (rationals), $\mathbb{Q}^c$ (complement of rationals), $\mathbb{R}$ (reals), $\mathbb{Z}$ (integers), $\mathbb{N}$ (naturals), $\emptyset$ (empty set), and $\mathbb{W}$ (whole numbers). 2. Key definitions and rules: - $A \subset B$ means set $A$ is a subset of set $B$. - $A \cap B$ is the intersection of sets $A$ and $B$ (elements common to both). - $A \cup B$ is the union of sets $A$ and $B$ (all elements in either set). - $\emptyset$ is the empty set with no elements. - $card(A)$ denotes the cardinality (size) of set $A$. 3. Given statements: - $\mathbb{Q} \subset \mathbb{Q}^c$ is false because $\mathbb{Q}$ and its complement $\mathbb{Q}^c$ are disjoint. - $\mathbb{R} \subset \mathbb{R}$ is true (a set is always a subset of itself). - $\mathbb{Z} \subset \mathbb{Q}$ is true since all integers are rational numbers. - $\emptyset \subset \emptyset$ is true (empty set is subset of itself). 4. Intersections: - $\mathbb{Q} \cap \mathbb{Q}^c = \emptyset$ since rationals and their complement have no common elements. - $\mathbb{Q} \cap \mathbb{N} = \mathbb{N}$ because natural numbers are a subset of rationals. - $\mathbb{W} \cap \emptyset = \emptyset$ since intersection with empty set is empty. - $\mathbb{Z} \cup \mathbb{Z} = \mathbb{Z}$ union with itself is itself. 5. Unions: - $\mathbb{R} \cup \mathbb{Q}^c = \mathbb{Q}^c$ is false; actually $\mathbb{R} \cup \mathbb{Q}^c = \mathbb{R}$ because $\mathbb{Q}^c$ is subset of $\mathbb{R}$. - $\mathbb{N} \cup \emptyset = \mathbb{N}$ union with empty set is itself. 6. Cardinalities: - $card(\mathbb{N}) = card(\mathbb{Z})$ means the natural numbers and integers have the same cardinality (countably infinite). Final answers: - $\mathbb{Q} \subset \mathbb{Q}^c$: False - $\mathbb{R} \subset \mathbb{R}$: True - $\mathbb{Z} \subset \mathbb{Q}$: True - $\emptyset \subset \emptyset$: True - $\mathbb{Q} \cap \mathbb{Q}^c = \emptyset$ - $\mathbb{Q} \cap \mathbb{N} = \mathbb{N}$ - $\mathbb{W} \cap \emptyset = \emptyset$ - $\mathbb{Z} \cup \mathbb{Z} = \mathbb{Z}$ - $\mathbb{R} \cup \mathbb{Q}^c = \mathbb{R}$ - $\mathbb{N} \cup \emptyset = \mathbb{N}$ - $card(\mathbb{N}) = card(\mathbb{Z})$ These illustrate basic set theory properties and cardinality concepts.