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        Psychology 406

        Assignment \#1


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1) Hays --- Chapter 1, \#14, \#26, \#32, \#34; Chapter 2, \#2

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2) Suppose we denote the collection of 26 small letters \{a, b, c,
etc.\} by the letter A.

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a) Define the set A by means of a property.

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b) Each subset of A must contain less than \rule{.5in}{.01in}
members.

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c) Which of the following expressions are correct?  If an expression
is incorrect, explain why.

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\{a,e,i,o,u\} $\subset$ A

\{a,e,i,o,u\} $\subseteq$ A

\{a,e,i,o,u\} $\in$ A

$\emptyset$ $\subseteq$ $\emptyset$

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Consider A to be the universal set.

d) Define the complement of the set of vowels \{a,e,i,o,u\} by means
of a property.

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e) Let B = \{a,e,i,o,u\}, C = \{a,b,c,d\}, D = \{a,c,d,e,i,o\}

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A $\cup$ B =

A $\cap$ C $\cap$ D =

B $-$ A =

$\bar{\mathrm{B}}$ $\cup$ C =

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f) Using the sets B, C, and D given above, show that

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i) B $\cup$ (C $\cap$ D) = (B $\cup$ C) $\cap$ (B $\cup$ D)

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ii) B $\cap$ (C $\cup$ D) = (B $\cap$ C) $\cup$ (B $\cap$ D)

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iii) $\overline{\mathrm{B} \cap \mathrm{C}}$ = $\bar{\mathrm{B}}$
$\cup$ $\bar{\mathrm{C}}$

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iv) $\overline{\mathrm{B} \cup \mathrm{C}}$ = $\bar{\mathrm{B}}$
$\cap$ $\bar{\mathrm{C}}$

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Also, verify the four relationships using Venn diagrams.

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g) Why is the expression B $\cup$ C $\cap$ D ambiguous.

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3) Suppose a fair coin (probability of heads is 1/2) is flipped
three times and the outcomes recorded as a unit, e.g., (H,T,H).

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a) List the members of the sample space $\mathcal{S}$.  What are the
elementary events?

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b) If all outcomes of the three flips are equally likely, what is
the probability associated with any one single elementary event?

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c) Let A be the set of outcomes containing exactly one head; B the
outcomes containing exactly two heads.

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i) P(A) =

ii) P(B) =

iii) P(A $\cup$ B) =

iv) P(A $\cap$ B) =

v) P($\bar{\mathrm{A}}$ $\cap$ B) =

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4) Stanley Student is typical of most graduate TAs, but has one
distinct disadvantage.  Because his wife is a nurse, the thermometer
they have is calibrated in Centigrade rather than Fahrenheit.  One
night, while his wife was working, their new baby started to cry ---
much more than usual.  Stanley decided to take her temperature and
after much more crying, succeeded in getting a reading of
37$^{\circ}$.  Unable to remember how to convert to Fahrenheit, he
reasoned as follows: I know that water boils at 100$^{\circ}$C and
212$^{\circ}$F; water freezes at 0$^{\circ}$C and 32$^{\circ}$F. So,
if I take 37\% of the difference between 212$^{\circ}$ and
32$^{\circ}$ (180) and add it to 32, I should have the Fahrenheit
reading:

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.37(180) + 32 = 98.6

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\noindent Stanley was satisfied that his child did not have a fever.
Was his reasoning correct? Explain.

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5) Suppose we assign to any set A the number of elements in the set.
For instance,

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\{a,b,d\} $\rightarrow$ 3

\{a\} $\rightarrow$ 1, etc.

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Discuss the highest type of scale that such a measurement procedure
might yield.







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