Theory of Positive Integers : Logic

Example Questions

Example Question #1 : Logic

over the domain

For all  which  is true?

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,

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,

Example Question #2 : Logic

Negate the following statement.

is a prime number.

is not a prime number

is an odd number

is a prime number

is an even number

is not a prime number

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,

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,

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.

is not a prime number.

Example Question #3 : Logic

Determine which statement is true giving the following information.

is a prime number  is odd

Explanation:

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

The first component of this statement is:

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:

is odd

This statement is false since .

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

Example Question #4 : Logic

over the domain

For all  which  is true?