Math 110-3.1 and 3.2 Fall 2026

Truth Value- Or v

It is raining or it is snowing p v q is true

if it’s raining

if it’s snowing or both

p v q if false

if it not raining or if it’s snowing

• The statement 𝑝 ∧ 𝑞 is true only when the individual statements
(p,q) are both true.
• The statement 𝑝 ∧ 𝑞 is false whenever either of the individual
statements (p,q) is false

I am tired and hungry p ^ q is true

I am tired but not hungry p ^ q is false

I am hungry but not tried p ^ q is false

Memorize The Truth Table for And ∧

EXAMPLE 3

Determine the truth value of ~𝑝 ∨ (𝑞 ∧∼ 𝑟), given that 𝑝 is false,
𝑞 is true, and 𝑟 is false.

~pv (q ^ ~ r)

~Fv (T ^ ~ f)

Tv (T^~ f)

Tv t = true

EXAMPLE 4

Determine the truth value of (𝑝 ∧ 𝑞) ∨ (~𝑝 ∧ ~𝑞), given that 𝑝 is false and 𝑞 is
false.

(p ^ q)v(~p^~q)

(F^F)v(`F^~F)

F V (T ^ T)

F V T=true

EXAMPLE 5

Determine the truth value of 𝑝 ∧ ~𝑟 ∨ [~(𝑝 ∧ 𝑟) ∨ 𝑞], given that 𝑝 is true, 𝑞 is
false, and 𝑟 is true.

(P^~r)

Math 3.3 and 3.4

Write the statement in “If 𝑝, then 𝑞” form.

(a) (Your vocabulary will expand) (if you read more.)

P Q

(NOTE: This is in 𝑞 if 𝑝 form)

Answer: If you read more, then your vocabulary will expand.

(b) (Your grade may improve) provided that (you spend more time studying.)

Q P
(NOTE: this is in 𝑞 provided that 𝑝 form)

Answer: If you spend more time studying, then your grade may improve.

P Q

(NOTE: This is in Every 𝑝 is a 𝑞 form)

Answer: If you are a U.A. student, then you are a member of the campus
community.

P Q
(d) All (squares) are (rectangles.) (NOTE: This is in All 𝑝‘s are 𝑞‘s)

Answer: If it is a square, then it is a rectangle.

Statements
Related to
the
Conditional

There are 3 statements related but
not necessarily equivalent to the
conditional 𝑝 → 𝑞:
1. The converse of the conditional is:
𝑞 → 𝑝
2. The inverse of the conditional is:
~𝑝 → ~𝑞
3. The contrapositive of the
conditional is: ~𝑞 → ~𝑝

Write the converse, inverse, and
contrapositive of the given statement:
If (it is Saturday), (then it is the
weekend.)

Converse 𝑞 → 𝑝: If it is the weekend,
then it is Saturday.

Inverse ~𝑝 → ~𝑞: If it is not
Saturday, then it is not the weekend.

Contrapositive ~𝑞 → ~𝑝: If it is not
the weekend, then it is not Saturday

Write the converse, inverse, and contrapositive of the given statement:

(I will spend less time on my phone) provided that (I read more.)

Q P
(NOTE: This is in 𝑞 provided that 𝑝 form, so let’s first rewrite it in 𝑝 → 𝑞 form:

If (I read more,) then (I will spend less time on my phone.)

P Q
Converse 𝑞 → 𝑝: If I spend less time on my phone, then I will read more.

Inverse ~𝑝 → ~𝑞: If I do not read more, then I will not spend less time on my phone.

Contrapositive ~𝑞 → ~𝑝: If I do not spend less time on my phone, then I will not read more.

Write the converse, inverse, and
contrapositive of the given
statement:

If (𝑥 = 7) then 𝑥 + 4 = 11

Converse 𝑞 → 𝑝:

Inverse ~𝑝 → ~𝑞:

Contrapositive ~𝑞 → ~𝑝:

Math 13.1 & 13.2- Measures of Central
Tendency and Dispersion

We will find the middle/center/typical value of a set of data, using 3 methods:
the mean, median, and mode. We will also discuss how “spread out” data is,
using the range, standard deviation, and variance.

MEAN

Definition: The mean of a given set
of data is the arithmetic average.

• To find the mean, add up all the data
values and then divide by the
number of data values.

Example: Six students received the
following grades on a test: 92, 84,
65, 76, 76, and 90. Find the mean of
the test scores.

92+84+55+76+76+90= 483/6 Is 80.5

MEDIAN

Definition: The median is the middle value
(if an odd number of data values), or the
mean of the two middle values (if an even
number of data values). The data must be in
numerical order (least to greatest or greatest
to least – doesn’t matter).

Example: For the set of test scores 92, 90,
84, 76, 65 the median is:

65,76,84,90,92= MEDIAN is 84

Example: For the set of test scores 92, 90,
84, 76, 76, 65 the median is:

65,76,76,84,90,92= 76+84/2 = 160/2 = 80

MODE

• Definition: The mode is the value that
occurs most often. There may be one
mode, more than one mode, or no
mode.

Example: Find the mode for the
following set of data: 92, 90, 84, 76,
76, 65 = The most is 76

Example: Find the mode for the
following set of data: 9.2, 9.0, 8.4, 7.6,
6.5 = NO MODE (DNE)

Example: Find the mode for the
following set of data: -9, -9, -8.4, -7.6,
-7.6, -6.5 The most is = -9 and -7.6

13.1 & 13.2

Weighted Mean

The weighted mean is used when not
all data values are equally important
(i.e., have different “weights”)
Formula for weighted mean =
𝑠𝑢𝑚 𝑜𝑓 (𝑣𝑎𝑙𝑢𝑒 ∗ 𝑤𝑒𝑖𝑔ℎ𝑡)
𝑡𝑜𝑡𝑎𝑙 𝑜𝑓 𝑤𝑒𝑖𝑔ℎ𝑡𝑠

Find mean of Alan’s test scores

80+84+76+96=336/4=84

15% (x4 = 60%) Tests: 80,84,76,96
20% Term Paper: 82
20% Final Exam: 92

Multiply data values by weight

84(0.6)+82(0.2)+92(0.2)/1

50.4+15.4+18.4=85.2

70(0.5)+90(0.3)+85(0.05)+0.15x=80

35+27+4.25+0.15x=80

~~66.25~~ +0.15x=80

~~66.25~~ -66.25

0.15/0.15=13.75/0.15 x=91.67

(a) Your GPA is a weighted mean, where
each course is weighted according to its
number of credits. Letter grades are given
the following values:
A = 4.00, A- = 3.67, B+ = 3.33, B = 3.00, MUTITPLY BY THE NUMBER OF CREDITS
B- = 2.67, C+ = 2.00, C = 2.00,
C- = 1.67, D+ = 1.33, D = 1.00, D- = 0.67,
F = 0.
With the following grades, what is your
GPA? Round to the nearest hundredth

3.67(3)+4.00(4)+3.33(4)+1.67(1)

Scores on a Biology Quiz- Frequency

2 1

4 2

6 7

7 12

8 12

Formula= Total let x=score of the 7th test

Numbers

6(77) tx =79

RANGE

Def: The range is the difference between
the largest and smallest values in a set of
data. So, subtract: largest data value –
smallest data value

Example: Six students received the
following grades on a test: 92, 84, 65, 76,
76, and 90. Find the range

92-65=27

Standard Deviation
• Def: Given a set of data, the standard
deviation measures the average deviation
from the mean, or how far from the mean
that most of the data lies.

Steps to Computing Standard Deviation
Step 1: Find the mean
Step 2: Find all deviations from the mean (meaning each data value minus the mean)
Step 3: Square the deviations and then add
Step 4: Divide the result from step 3 by 𝑛 − 1 (where n is the number of data
values)
NOTE: Step 4 finds the variance
Step 5: Take the square root of the variance
NOTE: This step finds the standard deviation

Example 6
• Six students received the following grades
on a test: 92, 84, 65, 76, 76, 90. Find the
range, variance, and standard deviation.

• Range: 92-65=27

• Variance/Standard Deviation:

Step 1: find mean

92+84+65+76+76+90/6=80.5

Example 6 Continued
Steps 2-4:

Step 5:

Find the range, the standard
deviation, and the variance for the
given data: 3,5,9,11. Round non-
integer results to the nearest tenth.
• Range:
• Variance/Standard Deviation:
Step 1: