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I am just starting a discrete math course at university and this is the first question on my first assignment. Sorry you will think it is extremely simple. Funny part is I can do most of the questions after this one.

Let $A = \{n \in \mathbb{N} : n \ \text{is odd} \ \}$,

$B = \{ x \in \mathbb{R} : 0 \le x < 100\}$,

$C = ℚ$, $D = \{3^n : n ∈ ℕ\}$

and $ℝ$ be the universal set.

Determine $A ∩ B ∩ D, (A \setminus D) ∩ B$

What I think:

$$A ∩ B ∩ D = \{x, n \mid x ∈ ℕ\text{ is odd and }3^n \mid n ∈ ℕ\}$$

$(A \setminus D) \cap B = {}$no idea

Michael Hardy
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Greg
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  • Be aware which definition of natural numbers is being used. If $\mathbb{N}={0,1,2,\ldots}$ then powers of $3$ are ${1,3,9,27,\ldots}$, but if $\mathbb{N}={1,2,3,\ldots}$ then the powers of $3$ are ${3,9,27,\ldots}$. – Marconius Aug 09 '15 at 23:31
  • @FlyByNight : You used \backslash where you should have used \setminus. Notice the difference: $A\backslash D$ versus $A\setminus D$. ${}\qquad{}$ – Michael Hardy Aug 09 '15 at 23:35
  • Hi @MichaelHardy, I didn't even know that \setminus was a "command". Thanks for point that out; I'll change it now. – Fly by Night Aug 09 '15 at 23:46
  • @MichaelHardy Oh, I see you already changed it... – Fly by Night Aug 09 '15 at 23:48

3 Answers3

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Don't get confused by the use of different pronumerals to mark what's in each set. In words, the sets are:

A: "The set of odd natural numbers"

B: "The set of real numbers between zero (inclusive) and 100 (exclusive)"

C: "The set of rational numbers"

D: "The set of powers of three"

So to write, for example, $A \cap B \cap D$, you want the set of all numbers that are (a) odd natural numbers and (b) real numbers from zero to 100 and (c) powers of three. Whereas for $(A \backslash D) \cap B$ you want all numbers that are odd natural numbers that are not powers of three, but that are between zero and 100. Can you write either of those in set notation?

ConMan
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Let

  • $A = \{x \mid x \in\Bbb N\text{ is odd}\} =\{x\in \Bbb N\mid \exists k\in\Bbb N: x=2k+1\}$,
  • $B = \{a \in\Bbb R \mid 0 \leq a < 100\}$,
  • $C = \Bbb Q$,
  • $D = \{3^n \mid n \in\Bbb N\} = \{x\in \Bbb N\mid \exists n\in\Bbb N: x= 3^n\} $
  • and $\Bbb R$ be the universal set.

Determine $A ∩ B ∩ D$,

Intersection means all set constructions must be satisfied.

The intersection between :odd integers, reals between $[0;100)$, and integer powers of 3 will be... integer powers of 3 that are odd and between $1$ and $81$ (inclusive).

In fact: $\{1,3,9,81\}$ Just those.

(A \ D) ∩ B

Set difference means the set construction of the divided set must not be satisfied.

Which means we have integers in the interval of $\{0..99\}$ that are odd but not integer powers of three.

Graham Kemp
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Well, in words:

The set $A \cap B \cap D$ is the set of "all elements of $\mathbb{R}$ that are odd natural numbers (in $A$), no less than 0 and strictly less than 100 (in $B$), and are natural powers of 3 (in $D$)".

First off, the only natural powers of 3 (elements of $D$) that are "no less than 0 and strictly less than 100" (in $B$) are _____.

Among these, _____ are odd natural numbers (in $A$), so we quickly see that

$$A \cap B \cap D = %\{1,3,9,27,81\}$$

(P.S. if a set is small, say 5 or fewer elements, then it is usually best just to list the elements explicitly).

For $(A \backslash D) \cap B$, we have "all elements of $\mathbb{R}$ that are odd natural numbers that are not natural powers of 3 (in $A \backslash D$), and are no less than 0 and strictly less than 100 (in $B$)".

Thus we have $(A \backslash D) \cap B = %\{1,2,5,7,11,\ldots,25,29,\ldots,79,83,\ldots,99\} = \{2n+1:0 \leq n \leq 49\}\backslash\{3,9,27,81\}$

727
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