집합과 명제 전체 문제

Let $S$, $T$, $X$, and $Y$ be subsets of some universal set. Assume that

• (i) $S \cup T \subseteq X \cup Y$; - (ii) $S \cap T = \emptyset$; and - (iii) $X \subseteq S$.

Using assumption (i), what conclusion(s) can be made if it is known that $a \in T$?

Let $S$, $T$, $X$, and $Y$ be subsets of some universal set. Assume that

• (i) $S \cup T \subseteq X \cup Y$; - (ii) $S \cap T = \emptyset$; and - (iii) $X \subseteq S$.

Using assumption (ii), what conclusion(s) can be made if it is known that $a \in T$?

Let $S$, $T$, $X$, and $Y$ be subsets of some universal set. Assume that

• (i) $S \cup T \subseteq X \cup Y$; - (ii) $S \cap T = \emptyset$; and - (iii) $X \subseteq S$.

Using all three assumptions, either prove that $T \subseteq Y$ or explain why it is not possible to do so.

Draw two general Venn diagrams for the sets $A$, $B$, and $C$. On one, shade the region that represents $\left( {A \cup B} \right) - C$, and on the other, shade the region that represents $\left( {A - C} \right) \cup \left( {B - C} \right)$. Explain carefully how you determined these regions and why they indicate that $\left( {A \cup B} \right) - C = \left( {A - C} \right) \cup \left( {B - C} \right)$.

Let $A$, $B$, and $C$ be subsets of some universal set $U$.

Draw two general Venn diagrams for the sets $A$, $B$, and $C$. On one, shade the region that represents $A - \left( {B \cup C} \right)$, and on the other, shade the region that represents $\left( {A - B} \right) \cap \left( {A - C} \right)$. Based on the Venn diagrams, make a conjecture about the relationship between the sets $A - \left( {B \cup C} \right)$ and $\left( {A - B} \right) \cap \left( {A - C} \right)$.

Let $A$, $B$, and $C$ be subsets of some universal set $U$.

Draw two general Venn diagrams for the sets $A$, $B$, and $C$. On one, shade the region that represents $A - \left( {B \cap C} \right)$, and on the other, shade the region that represents $\left( {A - B} \right) \cup \left( {A - C} \right)$. Based on the Venn diagrams, make a conjecture about the relationship between the sets $A - \left( {B \cap C} \right)$ and $\left( {A - B} \right) \cup \left( {A - C} \right)$.

Let $A$, $B$, and $C$ be subsets of some universal set $U$. Prove or disprove each of the following:

$\left( A \cap B \right) - C = \left( A - C \right) \cap \left( B - C \right)$

Let $A$, $B$, and $C$ be subsets of some universal set $U$. Prove or disprove each of the following:

$\left( A \cup B \right) - \left( A \cap B \right) = \left( A - B \right) \cup \left(B - A \right)$

Let $A$, $B$, and $C$ be subsets of some universal set $U$.

Draw two general Venn diagrams for the sets $A$, $B$, and $C$. On one, shade the region that represents $A - \left( {B - C} \right)$, and on the other, shade the region that represents $\left( {A - B} \right) - C$. Based on the Venn diagrams, make a conjecture about the relationship between the sets $A - \left( {B - C} \right)$ and $\left( {A - B} \right) - C$. (Are the two sets equal? If not, is one of the sets a subset of the other set?)

Let $A$, $B$, and $C$ be subsets of some universal set $U$.

Draw two general Venn diagrams for the sets $A$, $B$, and $C$. On one, shade the region that represents $A - \left( {B - C} \right)$, and on the other, shade the region that represents $\left( {A - B} \right) \cup \left( A - C^c \right)$. Based on the Venn diagrams, make a conjecture about the relationship between the sets $A - \left( {B - C} \right)$ and $\left( {A - B} \right) \cup \left( A - C^c \right)$. (Are the two sets equal? If not, is one of the sets a subset of the other set?)

Let $A$ and $B$ be subsets of some universal set $U$.

Prove that $A$ and $B - A$ are disjoint sets.

Let $A$ and $B$ be subsets of some universal set $U$.

Prove that $A \cup B = A \cup \left( {B - A} \right)$.

Let $A$ and $B$ be subsets of some universal set $U$.

Prove that $A - B$ and $A \cap B$ are disjoint sets.

Let $A$ and $B$ be subsets of some universal set $U$.

Prove that $A = \left( {A - B} \right) \cup \left( {A \cap B} \right)$.

Let $A$ and $B$ be subsets of some universal set $U$. Prove or disprove each of the following:

$A - \left(A \cap B^c \right) = A \cap B$

Let $A$ and $B$ be subsets of some universal set $U$. Prove or disprove each of the following:

$\left( A^c \cup B \right)^c \cap A = A - B$

Let $A$ and $B$ be subsets of some universal set $U$. Prove or disprove each of the following:

$\left( A \cup B \right) - A = B - A$

Let $A$ and $B$ be subsets of some universal set $U$. Prove or disprove each of the following:

$\left( A \cup B \right) - B = A - \left( A \cap B \right)$

Prove the following propositions:

If $A$, $B$, $C$, and $D$ are sets with $A \approx B$ and $C \approx D$ and if $A$ and $C$ are disjoint and $B$ and $D$ are disjoint, then $A \cup C \approx B \cup D$.