Answer
Here are the simplified representations of the sets:
**(a)** All integers less than or equal to 7:
\[ \{ \ldots, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7 \} \]
**(c₁)** All natural numbers less than 7:
\[ \{1, 2, 3, 4, 5, 6\} \]
**(c₂)** All integers greater than -3 and less than or equal to 1:
\[ \{-2, -1, 0, 1\} \]
**(g)** All non-negative natural numbers less than 12:
\[ \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\} \]
**(i)** All real numbers greater than -3 and less than or equal to 1:
\[ (-3,\, 1] \]
**(k)** All real numbers greater than 4.5:
\[ (4.5,\, \infty) \]
Solution
Sure, let's represent each of the given sets using appropriate mathematical notations.
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### **(a) \( \{x: x \leq 7 ;\ x \in \mathbb{Z}\} \)**
**Description:** This set includes all integers \( x \) that are less than or equal to 7.
**Representation:**
\[ \{ x \in \mathbb{Z} \mid x \leq 7 \} \]
Or enumerated:
\[ \{ \ldots, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7 \} \]
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### **(c₁) \( \{x: x < 7 ;\ x \in \mathbb{N}\} \)**
**Description:** This set contains all natural numbers \( x \) that are less than 7. Assuming \( \mathbb{N} \) represents the set \( \{1, 2, 3, \ldots\} \).
**Representation:**
\[ \{ x \in \mathbb{N} \mid x < 7 \} \]
Or enumerated:
\[ \{1, 2, 3, 4, 5, 6\} \]
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### **(c₂) \( \{x: -3 < x \leq 1 ;\ x \in \mathbb{Z}\} \)**
**Description:** This set includes all integers \( x \) that are greater than -3 and less than or equal to 1.
**Representation:**
\[ \{ x \in \mathbb{Z} \mid -3 < x \leq 1 \} \]
Or enumerated:
\[ \{-2, -1, 0, 1\} \]
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### **(g) \( \left\{x: x < 12 ;\ x \in \mathbb{N}_{0}\right\} \)**
**Description:** This set consists of all non-negative natural numbers \( x \) (where \( \mathbb{N}_{0} = \{0, 1, 2, \ldots\} \)) that are less than 12.
**Representation:**
\[ \{ x \in \mathbb{N}_{0} \mid x < 12 \} \]
Or enumerated:
\[ \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\} \]
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### **(i) \( \{x: -3 < x \leq 1 ;\ x \in \mathbb{R}\} \)**
**Description:** This set includes all real numbers \( x \) that are greater than -3 and less than or equal to 1.
**Representation:**
\[ \{ x \in \mathbb{R} \mid -3 < x \leq 1 \} \]
In interval notation:
\[ (-3,\, 1] \]
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### **(k) \( \left\{x: x > 4 \frac{1}{2} ;\ x \in \mathbb{R}\right\} \)**
**Description:** This set contains all real numbers \( x \) that are greater than \( 4.5 \) (since \( 4 \frac{1}{2} = 4.5 \)).
**Representation:**
\[ \{ x \in \mathbb{R} \mid x > 4.5 \} \]
In interval notation:
\[ (4.5,\, \infty) \]
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Each of these representations accurately describes the respective sets based on the given conditions.
Reviewed and approved by the UpStudy tutoring team
Explain
Simplify this solution
