To solve this problem first identify what the notation means and what exactly the question is asking.

\(\displaystyle \\n(A)=7\) means the number of elements in the set \(\displaystyle A\) is seven.

\(\displaystyle \\n(B)=12\) means the number of elements in the set \(\displaystyle B\) is twelve.

\(\displaystyle \\n(A\cap B)=\varnothing\) means the number of elements that exist in both \(\displaystyle A\) and \(\displaystyle B\) is the empty set, thus none of the elements that are in \(\displaystyle A\) are in \(\displaystyle B\).

Now the question asks to find, 

\(\displaystyle n(A\cup B)\) which means to find the number of unique elements that exist in \(\displaystyle A\) and in \(\displaystyle B\).

Since the intersection of the two sets is the empty set, that means all elements in \(\displaystyle A\) and in \(\displaystyle B\) are unique. 

Thus calculating the answer is as follows,

\(\displaystyle \\n(A\cupB)=n(A)+n(B)-n(A\cap B) \\=7+12-0\\=19\)

First, identify and understand the notation.

\(\displaystyle S\cap B\) means the intersection between the two sets. The intersection is when both situations overlap and are true.

\(\displaystyle \sim(\ \ \ \ )\) means "not the case"

Therefore  

\(\displaystyle \sim (S \cap B)\) means "It is not the case that the sun is shinning and I got sunburnt."

To solve this problem first identify what the notation means and what exactly the question is asking.

\(\displaystyle \\n(A)=7\) means the number of elements in the set \(\displaystyle A\) is seven.

\(\displaystyle \\n(B)=12\) means the number of elements in the set \(\displaystyle B\) is twelve.

\(\displaystyle \\n(A\cap B)=3\) means the number of elements that exist in both \(\displaystyle A\) and \(\displaystyle B\) are three.

Now the question asks to find, 

\(\displaystyle n(A\cup B)\) which means to find the number of unique elements that exist in \(\displaystyle A\) and in \(\displaystyle B\).

Since the intersection of the two sets is three,

Calculating the answer is as follows,

\(\displaystyle \\n(A\cupB)=n(A)+n(B)-n(A\cap B) \\=7+12-3\\=16\)

\(\displaystyle A\) and \(\displaystyle B\) are defined to be disjoint sets if their intersection has no elements. This occurs if and only if

\(\displaystyle c(A) + c(B) = c(A \cup B )\);

that is, if and only if the number of elements in the union is equal to the sum of the elements in the sets.

By substitution, we see that this is the statement

\(\displaystyle c(A) + c(B) = 142+108 = 250\)

\(\displaystyle c(A) + c(B) > C(A \cup B)\),

so \(\displaystyle A\) and \(\displaystyle B\) exist, and are not disjoint.

A set is well-defined if is defined in a way that makes it clear what elements are and are not in the set. The "ten best Presidents of the United States" is open to the judgment of whoever defines the set, so it is ambiguous which Presidents belong to that set.

A logical statement is a sentence that can be determined to be true or false. 7 is known to not be a composite number, so the sentence is known to be false; that makes it a valid example of a logical statement.

The statement "John is a carpenter and Jim is a taxi driver" is a compound statement, comprising two simple statements \(\displaystyle m , n\) connected with an "and". This statement is therefore

\(\displaystyle m\) and \(\displaystyle n\),

where

\(\displaystyle m :\) John is a carpenter

\(\displaystyle n:\)  Jim is a taxi driver.

The negation of a statement \(\displaystyle a\) is "not \(\displaystyle a\)"; the negation of "\(\displaystyle m\) and \(\displaystyle n\)" is

"Not ( \(\displaystyle m\) and \(\displaystyle n\))"

By DeMorgan's Law, this is equivalent to the statement

"(Not \(\displaystyle m\)) or (Not \(\displaystyle n\))"

Thus,"John is not a carpenter or Jim is not a taxi driver." is indeed the negation of the given statement.

The shaded portion of the Venn diagram is exactly the portion of the universal set \(\displaystyle U\) not in \(\displaystyle A\) - the complement of \(\displaystyle A\). Since \(\displaystyle A\) is the set of all people who listen to Band A, this complement is the set of all people who do not listen to Band A.

Let \(\displaystyle a\) and \(\displaystyle b\) be the simple statements:

\(\displaystyle a\): Andy is a Freemason

\(\displaystyle b\): Danny is not a Freemason.

The given conditional is therefore "If \(\displaystyle a\) then \(\displaystyle b\)".

The contrapositive of this conditional is defined to be "If not \(\displaystyle b\) then not \(\displaystyle a\)," which here is the statement "If Danny is a Freemason, then Andy is not a Freemason."

A set is well-defined if it can be determined with no ambiguity which elements are and are not in the set. In the case of \(\displaystyle P\), the word "smart" is ambiguous, since the definition can change according to who is deciding who is "smart" and who is not. \(\displaystyle P\) is therefore not a well-defined set.