Short explanation of mathematical logic and links to practice problems.

Here is a link to practice problems for mathematical logic.

There are four major logical operators we will be using.

AND, represented by the symbol ^.
OR, represented by the symbol v.
NOT, represented by the symbol ~
IMPLIES, represented by the symbol => (supposed to be an arrow, but no single symbol availabe to the blogger editor works.

There are only two values in logic, TRUE (represented by 1) and FALSE (represented by 0). A logical variable, often represented by the letters p, q and sometimes r, though other letters are legal, is an ordered list of 1s and 0s, known as a bitstring.

Here are the rules for AND, an operator that needs two variables.
1 ^ 1 = 1
1 ^ 0 = 0
0 ^ 1 = 0
0 ^ 0 = 0


In English, the simplest way to explain AND is to say p ^ q is only true if BOTH p and q are true, otherwise the operation produces a false outcome.


Here are the rules for OR, an operator that needs two variables.
1 v 1 = 1
1 v 0 = 1
0 v 1 = 1
0 v 0 = 0



In English, the simplest way to explain OR is to say p v q is true if EITHER p or q are true, and only false if both are false.


NOT is an operator that applies to a single variable.


~1 = 0
~0 = 1


NOT is like a switch. "NOT true" is the same as "false" and "NOT false" is the same as "true".


Mathematical logic is similar to set theory, where AND is like intersection, OR is like union and NOT is like the complement (or bar over the set).


IMPLIES is an operation in mathematical logic that doesn't have an analog to a basic set theory operator.  The rules are as follows.


1 => 1 = 1
1 => 0 = 0

0 => 1 = 1
0 => 0 = 1


If you have trouble remembering the rules for p => q, it works the same as ~p v q, where the NOT at the beginning only applies to the variable p. (The standard order of operations in logic is left to right, unless there are parentheses and ~ only applies to the variable immediately following it.)

A link to the stories of the five 19th Century British mathematicians who started mathematical logic.

Here is a link to the stories about Boole, Babbage, DeMorgan, Lovelace and Dodgson.

A link to all the posts dealing with binary and hexadecimal representation.

This link goes to three posts that deal with binary and hexadecimal representation. There are practice problems with answers in the comments.

Notes and practice for Venn diagrams and conditional probability

This link is to a post on my statistics blog about what we did today in the lab handout. Like posts here on the Math for Liberal Arts blog, there are problems at the end with answers in the comments.

Link to practice with set theory and cardinality.

Here's a link to some set theory problems.

Practice for contingency tables and time problems

_____________Male____Female__row totals
Facebook______55________________125
no Facebook __________20_____   ______
col. totals___90_____________________grand total

Answer the following probability problems from the completed contingency table. Write the answers first as fractions then rounded to the nearest thousandth.

The inclusion exclusion rule tells us p(A UNION B) = p(A) + p(B) - p(A INTERSECTION B) 

p(Facebook) = _________ ~= __________

p(Male) = _________ ~= __________

p(Male INTERSECTION Facebook) = _________ ~= __________

p(Male UNION Facebook) = _________ ~= __________

Time problems.

a) How many minutes in 3 days?

b) 560 hours is __________ days and ________ hours.


c) 8,086 seconds is ________ hours, ________ minutes and __________ seconds.

d) 8,086 hours is ________ weeks, ________days and __________ hours.

e) After a flu outbreak, a hospital has decided quarantine will last 66 hours. If it starts at 5:00 pm on Tuesday, give the day and time when it will be over.

Day: ________ time: ___________ a.m or p.m? __________

Answers in the comments.
 

Practice for regular polygon angles and pictures of regular tilings

Here is a link to practice problems from 2011 dealing with the angles of regular polygons.

Here are pictures of some mix and match tilings of the planes, which will be on the Tuesday homework.

Here are more regular polygon problems.

The formula for total degrees of the angles on an n-gon is 180(n-2)° or [180n-360]°.

 The formula for the measure of a regular angle is [180-360/n]°.

a) What is the sum of all the angles of a 20-gon?

b) What is the regular angle measure of a 20-gon?

c) What is the sum of all the angles of a 30-gon?

d) What is the regular angle measure of a 30-gon?


Answers to these questions in the comments.

Link to the story of the five computer scientists

Here is the link to the stories of John Von Neumann, Grace Hopper, Alan Turing, Edsger Dijkstra and Donald Knuth.

Order of operations practice

Using the rules of order of operations, find the values of the following expressions.

a) 2(3² - 5)

b) 2(3 - 5)² 

c) 2²(3 - 5)

d) 2 × 3² - 5 

e) (2 × 3)² - 5

f) 2(3² - 5²)

Answers in the comments.