Math / Discrete

QuestionFor each pair of statements, choose the one that is true. (a) $\{4\} \in\{4,5\}$ $4 \in\{4,5\}$ (b) $\{10,12,14\} \subseteq\{2,4,6,8, \ldots\}$ $\{10,12,14\} \in\{2,4,6,8, \ldots\}$ (c) $q \subseteq\{q, r\}$ $\{q\} \subseteq\{q, r\}$ $\{g\} \in\{e, g, h\}$ (d) $\{g\} \nsubseteq\{e, f, h\}$

Studdy Solution

STEP 1

1. We need to determine the truth value of each statement in the pairs.
2. The symbol $\in$ denotes membership, meaning an element is a member of a set.
3. The symbol $\subseteq$ denotes a subset, meaning all elements of one set are contained in another set.
4. The symbol $\nsubseteq$ denotes that a set is not a subset of another.

STEP 2

1. Analyze statement pair (a) to determine which is true.
2. Analyze statement pair (b) to determine which is true.
3. Analyze statement pair (c) to determine which is true.
4. Analyze statement (d) to determine its truth value.

STEP 3

Analyze statement pair (a):
- Statement: $\{4\} \in \{4,5\}$ - This statement is false because $\{4\}$ is a set, and $\{4,5\}$ contains the elements $4$ and $5$ , not the set $\{4\}$ .
- Statement: $4 \in \{4,5\}$ - This statement is true because $4$ is indeed an element of the set $\{4,5\}$ .
The true statement is: $4 \in \{4,5\}$ .

STEP 4

Analyze statement pair (b):
- Statement: $\{10,12,14\} \subseteq \{2,4,6,8,\ldots\}$ - This statement is true because $\{2,4,6,8,\ldots\}$ represents the set of all even numbers, and $10, 12, 14$ are all even numbers.
- Statement: $\{10,12,14\} \in \{2,4,6,8,\ldots\}$ - This statement is false because $\{10,12,14\}$ is a set, and $\{2,4,6,8,\ldots\}$ contains individual even numbers, not sets.
The true statement is: $\{10,12,14\} \subseteq \{2,4,6,8,\ldots\}$ .

STEP 5

Analyze statement pair (c):
- Statement: $q \subseteq \{q, r\}$ - This statement is false because $q$ is an element, not a set. The subset symbol $\subseteq$ is used between sets.
- Statement: $\{q\} \subseteq \{q, r\}$ - This statement is true because the set $\{q\}$ is indeed a subset of $\{q, r\}$ .
The true statement is: $\{q\} \subseteq \{q, r\}$ .

STEP 6

Analyze statement (d):
- Statement: $\{g\} \in \{e, g, h\}$ - This statement is false because $\{g\}$ is a set, and $\{e, g, h\}$ contains the elements $e, g, h$ , not the set $\{g\}$ .
- Statement: $\{g\} \nsubseteq \{e, f, h\}$ - This statement is true because the set $\{g\}$ is not a subset of $\{e, f, h\}$ since $g$ is not an element of $\{e, f, h\}$ .
The true statement is: $\{g\} \nsubseteq \{e, f, h\}$ .
The true statements are: (a) $4 \in \{4,5\}$ (b) $\{10,12,14\} \subseteq \{2,4,6,8,\ldots\}$ (c) $\{q\} \subseteq \{q, r\}$ (d) $\{g\} \nsubseteq \{e, f, h\}$

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