Logic, Sets, and Counting - Finite Mathematics

First, identify and understand the notation.

means the intersection between the two sets. The intersection is when both situations overlap and are true.

means "not the case"

Therefore

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.

means the number of elements in the set is seven.

means the number of elements in the set is twelve.

means the number of elements that exist in both and are three.

Now the question asks to find,

which means to find the number of unique elements that exist in and in .

Since the intersection of the two sets is three,

Calculating the answer is as follows,

The statement "John is a carpenter and Jim is a taxi driver" is a compound statement, comprising two simple statements connected with an "and". This statement is therefore

and ,

where

John is a carpenter

Jim is a taxi driver.

The negation of a statement is "not "; the negation of " and " is

"Not ( and )"

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

"(Not ) or (Not )"

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

and are defined to be disjoint sets if their intersection has no elements. This occurs if and only if

;

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

,

so and exist, and are not disjoint.

The shaded portion of the Venn diagram is exactly the portion of the universal set not in - the complement of . Since 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.

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

"Andy is a Freemason if and only if neither Benny nor Charlie is a Freemason."

is a biconditional statement. It can be rewritten by negating both parts of the statement as

"It is not true that Andy is a Freemason if and only if it is not true that neither Benny nor Charlie is a Freemason."

or, simplifying it a little,

"Andy is not a Freemason if and only if either Benny or Charlie is a Freemason."

Benny is a Freemason, making the second part of the latter biconditional statement true whether or not Charlie is a Freemason. It follows that Andy is not a Freemason. However, whether Charlie is a Freemason or not, his status is consistent with the biconditional.

We therefore know that Andy is not a Freemason, but we do not know whether or not Charlie is 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 , since, given any person in history, that person is/was or is/was not the President of the United States, is a well-defined set.

The selection of a license plate number can be seen as a series of independent events, as follows:

First, a prefix of one or two letters must be chosen. One way to look at this is that the prefix can be any one of 25 letters ("X" is excluded) followed by either any of 26 letters or a blank. By the multiplication principle, there are

possible prefixes.

The remaining characters must comprise five numeral; since there are no restrictions on the digits, by the multiplication principle, the number of possible numeral strings is

Applying the multiplication principle one more time, there will be

A logical statement is a sentence which is either true or false. "In ten years, the Prime Minister of the United Kingdom will be a woman" is a prediction. While its truth value will not be established for a while, it is either true or false, and it is therefore a logical statement.

Let and be the simple statements:

: Andy is a Freemason

: Danny is not a Freemason.

The given conditional is therefore "If then ".

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

Consider the second conditional "If Oscar is not a Freemason, then Nelson is not a Freemason.". It is known that Nelson is a Freemason, making the consequent of this conditional false. By a modus tollens argument, it follows that the antecedent is also false, and Oscar is a Freemason.

No conclusion can be drawn about Mickey, however. If Mickey is a Freemason, then by the first conditional, it follows that Nelson is a Freemason, which is already known; if Mickey is not a Freemason, no conclusion can be drawn that is inconsistent with what is known. Thus, either status is consistent with Nelson and Oscar both being Freemasons.

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 , the word "smart" is ambiguous, since the definition can change according to who is deciding who is "smart" and who is not. is therefore not a well-defined set.

A set is well-defined if each element can be identified with certainty as being or not being an element of the set. Since, for any value of , it can be clearly determined through substitution whether or not it is a solution of the given equation, is indeed well-defined.

The selection of a license plate number can be seen as a series of independent events, as follows:

First, the positions of the two letters is chosen. Since this is a choice of two positions out of six, without regard to order - the number of combinations of two from a set of six - the number of ways to choose these positions is .

Next, the two letters are chosen. Repetition is allowed, and there are 24 letters from which to choose, so, by the multiplication principle, there are ways to choose the letters.

Next, the four numerals are chosen. Repetition is allowed, and there are 10 digits from which to choose, so, by the multiplication principle, there are ways to choose the digits.

Applying the multiplication principle one more time, this gives us

different license plate numbers.

Let L stand for a letter and N stand for a numeral. One of two events will happen - the selection of a license plate with the pattern LNLNLNL, or the selection of a license plate with pattern NLNLNLN. These events are mutually exclusive, so we can count the number of ways to obtain them separately, then add.

There are no restrictions as to which letters or numerals can be chosen, or how many times each can be chosen, so the number of ways to choose a license plate number with the pattern LNLNLNL is

.

The number of ways to choose a license plate number with the pattern NLNLNLN is

.

Add these to get

,

the total number of license plates possible.

Call the hypothesis of the conditional, "", and call the conclusion, "". Then the given statement is the conditional "If then ."

The inverse of this conditional is the conditional "If (not ) then (not )/"- the conditional which negates the antecedent and the consequent,

"Not " is the negation of "", which is "It is not true that ", or, restated, "". Similarly, "Not " is the statement .

Thus, the inverse of the conditional is

"If , then ."

The negation of a statement can be stated as "It is not true that ." Therefore, the negation of the statement "" is the statement "It is not true that ." This can be restated as - not . Therefore, is not the negation of .