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Theory of Positive Integers : Theory of Positive Integers

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Example Questions

Example Question #1 : Sets

Let

$S:\begin{Bmatrix} -10,-8,-6,-4,-2,0,2,4 \end{Bmatrix}$ $\displaystyle S:\begin{Bmatrix} -10,-8,-6,-4,-2,0,2,4 \end{Bmatrix}$

if $\displaystyle B$ is some condition of $\displaystyle x$ such that it can be described as $\begin{Bmatrix} x\ \epsilon\ S: p(x) \end{Bmatrix}$ $\displaystyle \begin{Bmatrix} x\ \epsilon\ S: p(x) \end{Bmatrix}$ what is p(x) $\displaystyle p(x)$ when $B=\begin{Bmatrix} 2,4 \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} 2,4 \end{Bmatrix}$ ?

Possible Answers:

$B=\begin{Bmatrix} x\ \epsilon\ S: x<0 \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} x\ \epsilon\ S: x< 0 \end{Bmatrix}$

$B=\begin{Bmatrix} x\ \epsilon\ S: x\geq0 \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} x\ \epsilon\ S: x\geq0 \end{Bmatrix}$

$B=\begin{Bmatrix} x\ \epsilon\ S: x>0 \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} x\ \epsilon\ S: x>0 \end{Bmatrix}$

None of the answers

$B=\begin{Bmatrix} x\ \epsilon\ S: x\leq0 \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} x\ \epsilon\ S: x\leq0 \end{Bmatrix}$

Correct answer:

$B=\begin{Bmatrix} x\ \epsilon\ S: x>0 \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} x\ \epsilon\ S: x>0 \end{Bmatrix}$

Explanation:

First, identify what is given.

$S:\begin{Bmatrix} -10,-8,-6,-4,-2,0,2,4 \end{Bmatrix}$ $\displaystyle S:\begin{Bmatrix} -10,-8,-6,-4,-2,0,2,4 \end{Bmatrix}$

$B=\begin{Bmatrix} 2,4 \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} 2,4 \end{Bmatrix}$

and $\displaystyle B$ can be described in the following format

$B=\begin{Bmatrix} x\ \epsilon\ S: p(x) \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} x\ \epsilon\ S: p(x) \end{Bmatrix}$

Since $\displaystyle B$ contains the elements in $\displaystyle S$ that are greater than zero, p(x) $\displaystyle p(x)$ can be written as follows.

$B=\begin{Bmatrix} x\ \epsilon\ S: x>0 \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} x\ \epsilon\ S: x>0 \end{Bmatrix}$

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Example Question #2 : Sets

Let

$S:\begin{Bmatrix} -6,-4,-2,0,2,4 \end{Bmatrix}$ $\displaystyle S:\begin{Bmatrix} -6,-4,-2,0,2,4 \end{Bmatrix}$

if $\displaystyle B$ is some condition of $\displaystyle x$ such that it can be described as $\begin{Bmatrix} x\ \epsilon\ S: p(x) \end{Bmatrix}$ $\displaystyle \begin{Bmatrix} x\ \epsilon\ S: p(x) \end{Bmatrix}$ what is p(x) $\displaystyle p(x)$ when $B=\begin{Bmatrix} -6,-4,-2,0 \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} -6,-4,-2,0 \end{Bmatrix}$ ?

Possible Answers:

$B=\begin{Bmatrix} x\ \epsilon\ S: x\leq0 \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} x\ \epsilon\ S: x\leq0 \end{Bmatrix}$

$B=\begin{Bmatrix} x\ \epsilon\ S: x<0 \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} x\ \epsilon\ S: x< 0 \end{Bmatrix}$

$B=\begin{Bmatrix} x\ \epsilon\ S: x>0 \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} x\ \epsilon\ S: x>0 \end{Bmatrix}$

$B=\begin{Bmatrix} x\ \epsilon\ S: x\geq0 \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} x\ \epsilon\ S: x\geq0 \end{Bmatrix}$

None of the answers

Correct answer:

$B=\begin{Bmatrix} x\ \epsilon\ S: x\leq0 \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} x\ \epsilon\ S: x\leq0 \end{Bmatrix}$

Explanation:

First, identify what is given.

$S:\begin{Bmatrix} -6,-4,-2,0,2,4 \end{Bmatrix}$ $\displaystyle S:\begin{Bmatrix} -6,-4,-2,0,2,4 \end{Bmatrix}$

$B=\begin{Bmatrix} -6,-4,-2,0 \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} -6,-4,-2,0 \end{Bmatrix}$

and $\displaystyle B$ can be described in the following format

$B=\begin{Bmatrix} x\ \epsilon\ S: p(x) \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} x\ \epsilon\ S: p(x) \end{Bmatrix}$

Since $\displaystyle B$ contains the elements in $\displaystyle S$ that are less than or zero, p(x) $\displaystyle p(x)$ can be written as follows.

$B=\begin{Bmatrix} x\ \epsilon\ S: x\leq0 \end{Bmatrix}$ $\displaystyle B=\begin{Bmatrix} x\ \epsilon\ S: x\leq0 \end{Bmatrix}$

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Example Question #1 : Logic

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

For all $B\ \epsilon\ S$ $\displaystyle B\ \epsilon\ S$ which P(B) $\displaystyle P(B)$ is true?

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

Looking at what is given,

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

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

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

Thus the final solution is,

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

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Example Question #2 : Theory Of Positive Integers

Negate the following statement.

P: 17 $\displaystyle P: 17$ is a prime number.

Possible Answers:

$\sim P:17$ $\displaystyle \sim P:17$ is a prime number

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

$\sim P:17$ $\displaystyle \sim P:17$ is an odd number

$\sim P:17$ $\displaystyle \sim P:17$ is an even number

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

Correct answer:

$\sim P:17$ $\displaystyle \sim P:17$ is not a prime 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,

P: 17 $\displaystyle P: 17$ is a prime number.

contains to components.

Component one: P:17 $\displaystyle P:17$

Component two: "is a prime number"

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

$\sim P:17$ $\displaystyle \sim P:17$

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.

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

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Example Question #3 : Theory Of Positive Integers

Determine which statement is true giving the following information.

P: 17 $\displaystyle P: 17$ is a prime number Q:50 $\displaystyle Q:50$ is odd

Possible Answers:

$P\wedge Q$ $\displaystyle P\wedge Q$

$\sim (P\vee Q)$ $\displaystyle \sim (P\vee Q)$

$\sim P$ $\displaystyle \sim P$

$P\vee Q$ $\displaystyle P\vee Q$

None of the answers.

Correct answer:

$P\vee Q$ $\displaystyle P\vee Q$

Explanation:

To determine which statement is true first state what is known.

The first component of this statement is:

P: 17 $\displaystyle P: 17$ is a prime number

This is a true statement since only one and seventeen are factors of seventeen.

The second component of this statement is:

Q:50 $\displaystyle Q:50$ is odd

This statement is false since $50 \mod 2=0$ $\displaystyle 50 \mod 2=0$ .

Therefore, the only true statement is the one that uses the "or" operator because only one component is true.

Thus the correct answer is,

$P\vee Q$ $\displaystyle P\vee Q$

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Example Question #3 : Theory Of Positive Integers

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

For all $B\ \epsilon\ S$ $\displaystyle B\ \epsilon\ S$ which P(B) $\displaystyle P(B)$ is true?

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Looking at what is given,

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

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

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

Thus the final solution is,

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

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Example Question #1 : Theory Of Positive Integers

Which of the following is a property of a relation?

Possible Answers:

Symmetric Property

All are properties of a relation

Partition Property

Non-symmetric Property

Equivalency Property

Correct answer:

Symmetric Property

Explanation:

For a relation to exist there must be a non empty set present. If a non empty set is present then there are three relation properties.

These properties are:

I. Reflexive Property

$x\approx x$ $\displaystyle x\approx x$

II. Symmetric Property

$x\approx y\rightarrow y\approx x$ $\displaystyle x\approx y\rightarrow y\approx x$

III. Transitive Property

$x\approx y, y\approx z \rightarrow x\approx z$ $\displaystyle x\approx y, y\approx z \rightarrow x\approx z$

When all three properties represent a specific set, then that set is known to have an equivalence relation.

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Example Question #5 : Theory Of Positive Integers

What is an equivalency class?

Possible Answers:

$[x]=\begin{Bmatrix} y\ \epsilon\ A, -x\approx y \end{Bmatrix}$ $\displaystyle [x]=\begin{Bmatrix} y\ \epsilon\ A, -x\approx y \end{Bmatrix}$

$[x]=\begin{Bmatrix} y\ \epsilon\ A, x\approx y \end{Bmatrix}$ $\displaystyle [x]=\begin{Bmatrix} y\ \epsilon\ A, x\approx y \end{Bmatrix}$

$[x]=\begin{Bmatrix} y\ \epsilon\ A, x>y \end{Bmatrix}$ $\displaystyle [x]=\begin{Bmatrix} y\ \epsilon\ A, x>y \end{Bmatrix}$

$[x]=\begin{Bmatrix} y\ \epsilon\ A\end{Bmatrix}$ $\displaystyle [x]=\begin{Bmatrix} y\ \epsilon\ A\end{Bmatrix}$

$[x]=\begin{Bmatrix} x\approx y \end{Bmatrix}$ $\displaystyle [x]=\begin{Bmatrix} x\approx y \end{Bmatrix}$

Correct answer:

$[x]=\begin{Bmatrix} y\ \epsilon\ A, x\approx y \end{Bmatrix}$ $\displaystyle [x]=\begin{Bmatrix} y\ \epsilon\ A, x\approx y \end{Bmatrix}$

Explanation:

An equivalency class is a definitional term.

Suppose $\displaystyle A$ is a non empty set and $\approx$ $\displaystyle \approx$ is an equivalency relation on $\displaystyle A$ . Then $\displaystyle x$ belonging to $\displaystyle A$ is a set that holds all the elements that live in $\displaystyle A$ that are equivalent to $\displaystyle x$ .

In mathematical terms this looks as follows,

$[x]=\begin{Bmatrix} y\ \epsilon\ A, x\approx y \end{Bmatrix}$ $\displaystyle [x]=\begin{Bmatrix} y\ \epsilon\ A, x\approx y \end{Bmatrix}$

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Example Question #2 : Theory Of Positive Integers

Which of the following is a property of a relation?

Possible Answers:

All are relation properties

Equivalency Property

Non-symmetric Property

Associative Property

Reflexive Property

Correct answer:

Reflexive Property

Explanation:

For a relation to exist there must be a non empty set present. If a non empty set is present then there are three relation properties.

These properties are:

I. Reflexive Property

$x\approx x$ $\displaystyle x\approx x$

II. Symmetric Property

$x\approx y\rightarrow y\approx x$ $\displaystyle x\approx y\rightarrow y\approx x$

III. Transitive Property

$x\approx y, y\approx z \rightarrow x\approx z$ $\displaystyle x\approx y, y\approx z \rightarrow x\approx z$

When all three properties represent a specific set, then that set is known to have an equivalence relation.

Report an Error

Example Question #3 : Theory Of Positive Integers

Which of the following is a property of a relation?

Possible Answers:

Partition Property

All are properties of relations.

Transitive Property

Non-symmetric Property

Equivalency Property

Correct answer:

Transitive Property

Explanation:

For a relation to exist there must be a non empty set present. If a non empty set is present then there are three relation properties.

These properties are:

I. Reflexive Property

$x\approx x$ $\displaystyle x\approx x$

II. Symmetric Property

$x\approx y\rightarrow y\approx x$ $\displaystyle x\approx y\rightarrow y\approx x$

III. Transitive Property

$x\approx y, y\approx z \rightarrow x\approx z$ $\displaystyle x\approx y, y\approx z \rightarrow x\approx z$

When all three properties represent a specific set, then that set is known to have an equivalence relation.

Report an Error

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Theory of Positive Integers : Theory of Positive Integers

Study concepts, example questions & explanations for Theory of Positive Integers

All Theory of Positive Integers Resources

Example Questions

Example Question #1 : Sets

Example Question #2 : Sets

Example Question #1 : Logic

Example Question #2 : Theory Of Positive Integers

Example Question #3 : Theory Of Positive Integers

Example Question #3 : Theory Of Positive Integers

Example Question #1 : Theory Of Positive Integers

Example Question #5 : Theory Of Positive Integers

Example Question #2 : Theory Of Positive Integers

Example Question #3 : Theory Of Positive Integers

All Theory of Positive Integers Resources

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