집합의 뜻과 포함 관계

Let $A$, $B$, $C$, and $D$ be subsets of some universal set $U$. Are the following propositions true or false? Justify your conclusions.

If $A \subseteq B$ and $C \subseteq D$ and $A$ and $C$ are disjoint, then $B$ and $D$ are disjoint.

Let $A$, $B$, $C$, and $D$ be subsets of some universal set $U$. Are the following propositions true or false? Justify your conclusions.

If $A \subseteq B$ and $C \subseteq D$ and $B$ and $D$ are disjoint, then $A$ and $C$ are disjoint.

Let $A$, $B$, and $C$ be subsets of a universal set $U$. Prove:

If $A \subseteq B$, then $A \cap C \subseteq B \cap C$.

Let $A$, $B$, and $C$ be subsets of a universal set $U$. Prove:

If $A \subseteq B$, then $A \cup C \subseteq B \cup C$.

Let $A$, $B$, and $C$ be subsets of a universal set $U$. Are the following propositions true or false? Justify your conclusions.

If $A \cap C \subseteq B \cap C$, then $A \subseteq B$.

Let $A$, $B$, and $C$ be subsets of a universal set $U$. Are the following propositions true or false? Justify your conclusions.

If $A \cup C \subseteq B \cup C$, then $A \subseteq B$.

Let $A$, $B$, and $C$ be subsets of a universal set $U$. Are the following propositions true or false? Justify your conclusions.

If $A \cup C = B \cup C$, then $A = B$.

Let $A$, $B$, and $C$ be subsets of a universal set $U$. Are the following propositions true or false? Justify your conclusions.

If $A \cap C = B \cap C$, then $A = B$.

Let $A$, $B$, and $C$ be subsets of a universal set $U$. Are the following propositions true or false? Justify your conclusions.

If $A \cup C = B \cup C$ and $A \cap C = B \cap C$, then $A = B$.

Prove the following proposition:

For all sets

A

,

B

, and

C

that are subsets of some universal set, if

A \cap B = A \cap C

and

A^c \cap B = A^c \cap C

, then

B = C

.

Are the following biconditional statements true or false? Justify your conclusion. If a biconditional statement is found to be false, you should clearly determine if one of the conditional statements within it is true and provide a proof of this conditional statement.

For all subsets $A$ and $B$ of some universal set $U$, $A \subseteq B$ if and only if $A \cup B = B$.

Are the following biconditional statements true or false? Justify your conclusion. If a biconditional statement is found to be false, you should clearly determine if one of the conditional statements within it is true and provide a proof of this conditional statement.

For all subsets $A$ and $B$ of some universal set $U$, $A \subseteq B$ if and only if $A \cap B = A$.

Are the following biconditional statements true or false? Justify your conclusion. If a biconditional statement is found to be false, you should clearly determine if one of the conditional statements within it is true and provide a proof of this conditional statement.

For all subsets $A$, $B$, and $C$ of some universal set $U$, $A \subseteq B \cup C$ if and only if $A \subseteq B$ or $A \subseteq C$.

Are the following biconditional statements true or false? Justify your conclusion. If a biconditional statement is found to be false, you should clearly determine if one of the conditional statements within it is true and provide a proof of this conditional statement.

For all subsets $A$, $B$, and $C$ of some universal set $U$, $A \subseteq B \cap C$ if and only if $A \subseteq B$ and $A \subseteq C$.

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.