Finite Mathematics

The water temperatures in a certain lake over the course of the last seven days are as follows.

Calculate the mean water temperature.

$\text{Mean}\approx 74$

$\text{Mean}\approx 75$

$\text{Mean}\approx 73$

$\text{Mean}\approx 76$

$\text{Mean}\approx 71$

Explanation

To calculate the mean of the water temperatures first recall the formula for mean.

$\text{Mean}=\frac{\text{Sum of Water Temperatures}}{#\text{Days}}$

Now substitute in the values from the problem and solve.

$\text{Mean}=\frac{73+78+70+80+73+78+68}{7}$

$\text{Mean}=\frac{520}{7}$

$\text{Mean}\approx 74$

Consider the conditional statements:

"If Mickey is a Freemason, then Nelson is a Freemason."

"If Oscar is not a Freemason, then Nelson is not a Freemason."

Nelson is a Freemason. What can be concluded about whether or not Mickey and Oscar are Freemasons?

Oscar is a Freemason; no conclusion can be drawn about Mickey.

Mickey is a Freemason; no conclusion can be drawn about Oscar.

Mickey and Oscar are both Freemasons.

No conclusion can be drawn about either Mickey or Oscar.

Mickey is a Freemason; Oscar is not a Freemason.

Explanation

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.

$A = \left { 1, 2, 3, 4,5 \right }$

$B = \left { \textup{a, b, c, d} \right }$

True or false: $A \times B$ has more than one million subsets.

True

False

Explanation

$A \times B$ denotes the Cartesian product of and , the set of all ordered pairs comprising an element of followed by an element of . The number of elements in $A \times B$ is equal to the product of the numbers of elements in and :

$n(A \times B)= n(A)n(B)= 5 (4)= 20$ .

The number of subsets of a set with elements is $2^{N}$ , so $A \times B$ has $2^{20} = 1,048,576$ subsets total. The statement is true.

Let be the set of all persons who died in 1941.

True or false: is a well-defined set.

True

False

Explanation

A set is well-defined if it is clear which elements are in the set and which elements are not. There is a clear distinction between the people who died in 1941 and people who did not, so is well-defined.