집합의 뜻과 포함 관계

One of the properties of real numbers is the so-called Law of Trichotomy,

which states that if $a, b \in \mathbb{R}$, then exactly one of the following is true:

• $a \lt b$; - $a = b$; - $a > b$.

Is the following proposition concerning sets true or false? Either provide a proof that it is true or a counterexample showing it is false. If $A$ and $B$ are subsets of some universal set, then exactly one of the following is true:

• $A \subseteq B$; - $A = B$; - $B \subseteq A$.

Is the following proposition true or false? Justify your conclusion with a proof or a counterexample.

For all sets

A

and

B

that are subsets of some universal set

U

, the sets

A \cap B

and

A - B

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 $A$ 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 $U$. 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 $A = B$.

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 $A \subseteq C$.