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Which of the following sets is defined by [tex]\{x \in \mathbb{Z} : -3\ \textless \ x \leq 2\}[/tex]?

A. [tex]\{0,1,2,3\}[/tex]

B. [tex]\{-3,-2,-1\}[/tex]

C. [tex]\{-2,-1,0,1,2\}[/tex]

D. [tex]\{-3,-2,-1,0,1,2\}[/tex]


Sagot :

To solve the problem, we need to determine which set corresponds to the condition \(-3 < x \leq 2\), where \( x \) is an integer (i.e., \( x \in \mathbb{Z} \)).

Let's break down the condition step-by-step:

1. Understand the inequality:
- The condition \(-3 < x \leq 2\) indicates that \( x \) must be greater than \(-3\) but less than or equal to \( 2 \).

2. Determine the range of integers that satisfy the inequality:
- The integers greater than \(-3\) and less than or equal to \( 2 \) can be listed as follows:
[tex]\[ \{-2, -1, 0, 1, 2\} \][/tex]
- Note that \(-3\) is not included because the inequality is strict (i.e., \( x \) must be strictly greater than \(-3\)).
- The integer \( 2 \) is included because the inequality allows values that are less than or equal to \( 2 \) (i.e., \( \leq 2 \)).

3. Compare with the given options:
- Option (A) is \(\{0, 1, 2, 3\}\), which includes \( 3 \), an integer not within our defined range.
- Option (B) is \(\{-3, -2, -1\}\), which includes \(-3\), an integer not within our defined range.
- Option (C) is \(\{-2, -1, 0, 1, 2\}\), which perfectly matches our defined range of integers.
- Option (D) is \(\{-3, -2, -1, 0, 1, 2\}\), which includes \(-3\), an integer not within our defined range.

Hence, the correct set that represents the condition \(-3 < x \leq 2\) is:

[tex]\[ \boxed{3} \][/tex] (which corresponds to Option (C) [tex]\(\{-2, -1, 0, 1, 2\}\)[/tex]).