Answered

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22. If for each natural number [tex]n[/tex], [tex]s_n=\{100, 101, \ldots, 100+n\}[/tex], then which one of the following is not true?

A. [tex]s_n \cup s_{2n} = s_{2n}[/tex]

B. If [tex]n \ \textgreater \ m[/tex], then [tex]s_n \cap s_m = s_n[/tex]

C. [tex]s_{1000} = \{100, 101, \ldots, 1100\}[/tex]

D. [tex]s_{60} - s_{30} = \{131, 132, \ldots, 160\}[/tex]

Sagot :

GinnyAnswer
Let's start by analyzing each option step-by-step to determine which one is not true.

### Option A: [tex]\( s_n \cup s_{2n} = s_{2n} \)[/tex]

Set [tex]\( s_n = \{100, 101, \ldots, 100 + n\} \)[/tex] and [tex]\( s_{2n} = \{100, 101, \ldots, 100 + 2n\} \)[/tex].

The union [tex]\( s_n \cup s_{2n} \)[/tex] combines all elements from both sets:
- Elements of [tex]\( s_n \)[/tex] are from 100 to [tex]\( 100 + n \)[/tex].
- Elements of [tex]\( s_{2n} \)[/tex] are from 100 to [tex]\( 100 + 2n \)[/tex].

Since [tex]\( 100 + 2n \)[/tex] is always greater than [tex]\( 100 + n \)[/tex], the union of these sets will be exactly [tex]\( s_{2n} \)[/tex], i.e., [tex]\( s_n \cup s_{2n} = s_{2n} \)[/tex].

Thus, Option A is true.

### Option B: If [tex]\( n > m \)[/tex], then [tex]\( s_n \cap s_m = s_n \)[/tex]

Suppose [tex]\( n > m \)[/tex]. Then:
- [tex]\( s_n = \{100, 101, \ldots, 100 + n \} \)[/tex]
- [tex]\( s_m = \{100, 101, \ldots, 100 + m \} \)[/tex]

Since [tex]\( n > m \)[/tex], all elements of [tex]\( s_m \)[/tex] are also in [tex]\( s_n \)[/tex]. Therefore, the intersection [tex]\( s_n \cap s_m \)[/tex] contains exactly the elements of [tex]\( s_m \)[/tex] only, not [tex]\( s_n \)[/tex].

Thus, Option B is false.

### Option C: [tex]\( s_{1000} = \{100, 101, \ldots, 1100\} \)[/tex]

- [tex]\( s_{1000} = \{100, 101, \ldots, 100 + 1000\} = \{100, 101, \ldots, 1100\} \)[/tex]

This statement is directly equal to the definition of [tex]\( s_{1000} \)[/tex].

Thus, Option C is true.

### Option D: [tex]\( s_{60} - s_{30} = \{131, 132, \ldots, 160\} \)[/tex]

- [tex]\( s_{60} = \{100, 101, \ldots, 160\} \)[/tex]
- [tex]\( s_{30} = \{100, 101, \ldots, 130\} \)[/tex]

The difference [tex]\( s_{60} - s_{30} \)[/tex] removes elements that are in [tex]\( s_{30} \)[/tex] from [tex]\( s_{60} \)[/tex], leaving:
- Elements in [tex]\( s_{60} \)[/tex] but not in [tex]\( s_{30} \)[/tex] are [tex]\( 131, 132, \ldots, 160 \)[/tex].

Thus, Option D is true.

### Conclusion

By analyzing the provided options, the correct answer (the statement that is not true) is:

[tex]\[ \boxed{\text{B}} \][/tex]
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