Math / Discrete

QuestionWhich of the following statements are FALSE? $2 \in A \cup B$ implies that if $2 \notin A$ then $2 \in B$ $\{3\} \subseteq A-B$ and $\{2\} \subseteq B$ implies that $\{2,3\} \subseteq A \cup B$ . $\{2,3\} \subseteq A$ implies that $2 \in A$ and $3 \in A$ . $A \cap B \supseteq\{2,3\}$ implies that $\{2,3\} \subseteq A$ and $\{2,3\} \subseteq B$ . $\{2\} \in A$ and $\{3\} \in A$ implies that $\{2,3\} \subseteq A$ .

Studdy Solution

STEP 1

1. We are given a set of logical statements involving sets $A$ and $B$ .
2. We need to determine which of these statements are false.
3. Understanding the meaning of set operations such as union ( $\cup$ ), intersection ( $\cap$ ), set difference ( $-$ ), and subset ( $\subseteq$ ) is crucial.

STEP 2

1. Analyze the first statement.
2. Analyze the second statement.
3. Analyze the third statement.
4. Analyze the fourth statement.
5. Analyze the fifth statement.

STEP 3

Analyze the statement: $2 \in A \cup B$ implies that if $2 \notin A$ then $2 \in B$ .
- The statement $2 \in A \cup B$ means that $2$ is in either $A$ , $B$ , or both. - The implication $2 \notin A \Rightarrow 2 \in B$ is logically equivalent to saying if $2$ is not in $A$ , then it must be in $B$ for the union condition to hold true. - This statement is TRUE.

STEP 4

Analyze the statement: $\{3\} \subseteq A-B$ and $\{2\} \subseteq B$ implies that $\{2,3\} \subseteq A \cup B$ .
- $\{3\} \subseteq A-B$ means that $3$ is in $A$ but not in $B$ . - $\{2\} \subseteq B$ means that $2$ is in $B$ . - Therefore, $3$ is in $A$ and $2$ is in $B$ , which means both $2$ and $3$ are in $A \cup B$ . - This statement is TRUE.

STEP 5

Analyze the statement: $\{2,3\} \subseteq A$ implies that $2 \in A$ and $3 \in A$ .
- $\{2,3\} \subseteq A$ means both elements $2$ and $3$ are in $A$ . - Therefore, $2 \in A$ and $3 \in A$ is a direct consequence. - This statement is TRUE.

STEP 6

Analyze the statement: $A \cap B \supseteq \{2,3\}$ implies that $\{2,3\} \subseteq A$ and $\{2,3\} \subseteq B$ .
- $A \cap B \supseteq \{2,3\}$ means that both $2$ and $3$ are in the intersection of $A$ and $B$ . - This implies that both $2$ and $3$ are in $A$ and also in $B$ . - This statement is TRUE.

STEP 7

Analyze the statement: $\{2\} \in A$ and $\{3\} \in A$ implies that $\{2,3\} \subseteq A$ .
- $\{2\} \in A$ means the set containing $2$ is an element of $A$ , not that $2$ itself is an element of $A$ . - Similarly, $\{3\} \in A$ means the set containing $3$ is an element of $A$ . - This does not imply that $2$ and $3$ themselves are elements of $A$ . - This statement is FALSE.
The FALSE statement is: $\{2\} \in A$ and $\{3\} \in A$ implies that $\{2,3\} \subseteq A$ .

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