Theory of Positive Integers - Logic

$B\subseteq \begin{Bmatrix} {3,4,7,9,11} \end{Bmatrix}$ over the domain $S=P(\begin{Bmatrix} {4,7,10,12} \end{Bmatrix})$

For all $B\ \epsilon\ S$ which is true?

$B\ \epsilon\ \begin{Bmatrix} \O, 4,7 \end{Bmatrix}$

$B\ \epsilon\ \begin{Bmatrix} 4,7 \end{Bmatrix}$

$B\ \epsilon\ \begin{Bmatrix} 4,7,11 \end{Bmatrix}$

$B\ \epsilon\ \begin{Bmatrix} 3,7,10 \end{Bmatrix}$

$B\ \epsilon\ \begin{Bmatrix} 4,12 \end{Bmatrix}$

Explanation

This question is giving a subset who lives in the domain and it is asking for the partition or group of elements that live in both and .

Looking at what is given,

$B\subseteq \begin{Bmatrix} {3,4,7,9,11} \end{Bmatrix}$

$S=P(\begin{Bmatrix} {4,7,10,12} \end{Bmatrix})$

it is seen that both four and seven live in and therefore both these elements will be in the partition of . Another element that also exists in both sets is the empty set.

Thus the final solution is,

$B\ \epsilon\ \begin{Bmatrix} \O, 4,7 \end{Bmatrix}$

Negate the following statement.

is a prime number.

$\sim P:17$ is not a prime number

$\sim P:17$ is a prime number

is not a prime number

$\sim P:17$ is an even number

$\sim P:17$ is an odd number

Explanation

Negating a statement means to take the opposite of it.

To negate a statement completely, each component of the statement needs to be negated.

The given statement,

is a prime number.

contains to components.

Component one:

Component two: "is a prime number"

To negate component one, simply take the compliment of it. In mathematical terms this looks as follows,

$\sim P:17$

To negate component two, simply add a "not" before the phrase "a prime number".

Now, combine these two components back together for the complete negation.

$\sim P:17$ is not a prime number.

$B\subseteq \begin{Bmatrix} {2,10,11,23} \end{Bmatrix}$ over the domain $S=P(\begin{Bmatrix} {10,20,30} \end{Bmatrix})$

For all $B\ \epsilon\ S$ which is true?

$B\ \epsilon\ \begin{Bmatrix} \O, 10 \end{Bmatrix}$

$B\ \epsilon\ \begin{Bmatrix} 10 \end{Bmatrix}$

$B\ \epsilon\ \begin{Bmatrix} \O, 10,20 \end{Bmatrix}$

$B\ \epsilon\ \begin{Bmatrix} 10,23 \end{Bmatrix}$

$B\ \epsilon\ \begin{Bmatrix} \O \end{Bmatrix}$

Explanation

This question is giving a subset who lives in the domain and it is asking for the partition or group of elements that live in both and .

Looking at what is given,

$B\subseteq \begin{Bmatrix} {2,10,11,23} \end{Bmatrix}$

$S=P(\begin{Bmatrix} {10,20,30} \end{Bmatrix})$

it is seen that only ten lives in and therefore both these elements will be in the partition of . Another element that also exists in both sets is the empty set.

Thus the final solution is,

$B\ \epsilon\ \begin{Bmatrix} \O, 10 \end{Bmatrix}$