Mathematical Reasoning
By Michael Sakowski
Table of Contents
Unit 1 – Problem Solving and Statements, Negations, and Quantified Statements - Page 3
Unit 2 – Logic Statements - Page 10
Unit 3 – Truth Tables for Conditional & Biconditional & Equivalent Statements and DeMorgan's Laws – Page 15
Unit 4 – Arguments & Truth Tables and Euler Diagrams Applied to False Advertising – Page 20
Unit 5 – Fundamental Counting Principle and Permutations – Page 27
Unit 6 – Combinations and Fundamentals of Probability – Page 30
Unit 7 – Probability Using Counting Methods and Events Involving Not/Or & Odds, Casinos and Law of Large Numbers – Page 35
Unit 8 – Unit 8 - Events Involving And, Conditional Probability and Expected Value – Page 41
Unit 9 – Sampling, Frequency Distributions, Graphs and Measurements of Central Tendency – Page 45
Unit 10 – Measures of Dispersion and The Normal Distribution – Page 51
Unit 11 – Simple & Compound Interest and Annuities – Page 60
Unit 12 – Loans and Loan Payments – Page 66
Mathematical Reasoning
Unit 1 - Problem Solving and Statements, Negations, and Quantified Statements
Problem-Solving Strategies
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"Ninety Percent of All Mental Errors are in Your Head!" Yogi Berra
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Webster’s definition of “a problem” is
1. A question proposed for solution
2. A perplexing or difficult matter, person, etc.
A problem may usually be solved by employing the following 4 steps:
1. Identify what the real problem is. Remember the 3 R’s : reread, rephrase, and rewrite. Identify what is unknown. Draw a sketch if applicable.
2. Devise a plan for solution. Possibilities for math problems include:
Trial and Error
Make a table of values and look for a pattern
Use an algebraic equation
Obtain all needed formulas
(In the real world, its the solution that counts, not the method)
3. Carry out the plan.
4. Check your results. Do they make sense?
Example: A rectangle has an area of 15 square feet. Its length is 4 more than 4 times its width. What are the dimensions of this rectangle?
1. Identify what the real problem is. Remember the 3 R’s : reread, rephrase, and rewrite. Identify what is unknown. Draw a sketch if applicable.
We want to find length and width. We know that L = 4 + 4W where L represents length and W represents width. We also know that 15 = L*W since the area of a rectangle is equal to the product of length and width. From this information we need to find the values of L and W.
2. Devise a plan for solution. Possibilities for math problems include:
We need to find L and W so that both 15=L*W and L = 4 + 4W are both true. We can solve this by using trial and error. We can also solve algebraically.
3. Carry out the plan.
By trial and error, we pick a value for W, calculate L, and then see if the product is equal to 15.
|W |L (equals 4 + 4*W) |L*W = Area |
|1 |8 = 4 +4*1 |8 (too little) |
|2 |12 = 4 + 4*2 |24 (too much) |
|1.5 |10 = 4 + 4*1.5 |15 (just right!) |
| | | |
We could also solve this algebraically. We solve the system
L = 4 + 4W, L*W = 15 by substituting 4 + 4W into the 2nd equation to get
(4 + 4W)*W = 15
Multiply out to get 4W + 4W2 = 15 and rearrange algebraically to get 4W2 + 4W - 15 = 0
Factor this as (2W - 3)(2W + 5) = 0 and let each factor equal zero to get
2W - 3 = 0, 2W+5 = 0 with solutions W= 3/2 = 1.5 and W =-5/2. We only keep the positive solution of W=1.5 ft since a width can not be negative.
4. Check your results. Do they make sense?
For a width of 1.5 ft and a length of 10 ft, the length is indeed 4 more than 4 times the width and the product of width and length is 15 sq ft. Certainly the negative width would not make sense!
Final Exam Question: Solve a problem with the 4 step method of problem solving
Example: A small business purchases 35 widgets at $4 per dozen and sells them for 50 cents each. What is the net profit. Assume that widgets must be purchased in lots of a dozen.
Solution: Identify the problem: Find the net profit. Devise a Plan: Find total net cost and total gross profit. Subtract net cost from gross profit. Carry out the Plan: In this case, 3 dozen (36) widgets must be purchased at a cost of 3x4 = $12. They are then sold for $0.50 each, resulting in a gross profit of $0.50x35 = $17.50 . The NET profit is then $17.50 - $12 = $5.50 . Check Results: The results check and the answer seems reasonable.
Example: In the figure below, how many triangles are there?
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Solution: Identify Problem: Count the triangles, ALL of them. Devise a Plan: Count all triangles, including those made up of 2 or more triangles. Carry Out Plan: There are 3 smaller triangle (no lines through middle), 4 with a line through them, and 1 big one with two lines passing through the middle. This makes 8 total. Check: Results seem reasonable.
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Logic Statements
Statements, Negations, and Quantified Statements
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"When you come to a fork in the road, take it." Yogi Berra
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Introduction to Logic
Logic is used by. . . (Have Students Guess and then give examples of each)
Mathematicians - In proving theorems
Computer Scientists - If-then statements
Lawyers - In proving guilt or innocence
Philosophers - In analyzing truth and fallacies
Politicians - In fooling the public with errant reasoning
Statements and Symbols
A logic statement is a declarative sentence that is either true or false (but not true and false) .
Example: Which of the following is a logic statement?
A horse is not a cow.
A cow is a horse.
Don’t have a cow!
All non-zero even integers are divisible by 2.
Here are the answers:
A horse is not a cow IS a statement that is true
A cow is a horse IS a statement that happens to be false, but it still is a statement.
Don’t have a cow! IS NOT a statement because it is neither true nor false.
All non-zero even integers are divisible by 2 is a statement that is true.
Negations
The negation of a statement results in the opposite truth value of the statement.
Example: What is the negation of “The car is blue”? What is the symbolic form of the negation if "The car is blue" is represented by they symbol "C"? Note that we use ~ to represent the negation.
Answer: “The car is not blue”. Another equivalent negation would be “It is not the case that the car is blue”.
The symbol ~ is used to indicate the negation. If the symbol C represents the statement “The car is blue”, “The car is not blue” is represented as ~C.
Example: If S = "Sam is sneaky" and M= "Mary is merry", express each of the following with symbols.
A) It is not the case that Sam is not Sneaky.
B) It is the case that Mary is merry.
Solutions:
A) It is not the case that Sam is not Sneaky. would be represented by ~(~S), which chould be simplified to "S".
B) It is the case that Mary is merry. would be represented by simply writing "M".
Paradoxes
A paradox is a statement, that when assumed true, contradicts itself and also if it is assumed false, it contradicts itself.
Example: Is the sentence "This sentence is a lie" a paradox?
Answer: Yes, it is a paradox.
If true, then it implies it is a lie which would then make it not true. If false, then the sentence itself is a lie, then the implication is that it is not a lie and thus true, another contradiction!
Read about the Unexpected Hanging Paradox at and The Liars Paradox at 's_paradox
Oxymorons
Oxymorons are similar to paradoxes. They consist of a figure of speech in which contradictory terms appear together in the same sentence or phrase. For example “jumbo shrimp”, “minor crisis”, and “old news”. But we do not see the conflicting truth values as we do with paradoxes.
Quantifiers
The words “all, each, every, no, some, there exists, at least one, and none” are called quantifiers. Quantifiers how many of a particular situation exist.
Examples of statements with quantifiers below:
All people are human
Each person is a human (means the same as All people are human)
Every person is a human
No person is a human
None of the people are human
Some people are human (means the same as At least one person is a human)
There exists a person that is human (means the same as Some people are human)
At least one person is a human (means the same as Some people are human)
Negating Quantifiers
The negation of SOME DO is NONE DO or equivalently, ALL DO NOT.
The negation of ALL DO is SOME DON’T
SOME means AT LEAST ONE
Example: Write the statement "All journalists are writers" in an equivalent way. Also write the negation of this statement.
An equivalent way to write this is: "There are no journalists that are not writers". We can illustrate this with the diagram below where the set of all journalists would reside within the smaller circle and the set of all writers would reside within the larger outer circle. We can see that some writers may (like myself) possibly not be journalists since they could reside within the writers' circle but outside the journalists' circle.
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The negation of "All journalists are writers" would be "Some journalists are not writers". Note that if it is TRUE that "All journalists are writers" then it is automatically false that "Some journalists are not writers" - I point this out not to confuse you but to stress the point that a statement and its negation always have opposite truth values.
Example: Write the statement "Some movies are comedies" in an equivalent way. Also write the negation of this statement.
An equivalent way to write this is: "At least one movie is a comedy". We can illustrate this with the diagram below where the set of all comedies intersects the set of movies and there is at least one comedy that is within the movie circle. We can see that some comedies may possibly not be movies since they could reside outside the movie circle.
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The negation would be "All movies are not comedies". This situation is shown in fig 3 below. Here the 2 circles share no common ground. Again, both the original statement and its negation can not both have the same truth value - if one is true, the other must be false.
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Final Exam Question: Be able to write the negation of a statement that involve quantifiers like "all", "some", "every", etc.
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Unit 2 - Compound Statements & Connectives and Truth Tables for Negation, Conjunction, & Disjunction
Compound Statements and Connectives
Connectives
Compound statements consist of two or more statements connected with what are called “connectives”.
Example: “The sky is clear and it is raining” is a statement consisting of two statements connected with the word and.
Other connectives include “or”, “not”, “if ...then”, “if and only if”
More Examples:
I have a cold or I have hay fever.
If it is cold out, then I will need a jacket.
She was not happy and she was not poor.
And & Or Symbols used in Logic
Λ means “AND”. Statements with Λ in them are known as CONJUNCTIONS.
V means “OR”. Statements with V in them are known as DISJUNCTIONS.
So the statement I have a cold or I have hay fever could be written as P V Q where P = "I have a cold",
Q = "I have hay fever".
The statement I DON'T have a cold and I DON'T have hay fever could be written as ~P Λ ~Q .
If-Then Statements
If-then statement connect 2 statements together with "if" and "then" in the form IF P, THEN Q.
So the statement "IF I have a cold , THEN I feel sick" could be written as IF P, THEN Q where P = "I have a cold" and Q = "I feel sick".
Symbolic Form of IF A, THEN B
IF A, THEN B is written symbolically as A → B, So IF P, THEN Q where P = "I have a cold" and Q = "I feel sick" is written as P → Q.
Combinations of IF-THEN, AND, & OR
We can combine connectives as shown in the following example:
~A → (B V C) where
A = "I'm a monkey's uncle", B = "I'm a math teacher", & C = "I am a math student".
This statement would read "If I am not a monkey's uncle, then I am either a math teacher or a math student".
Biconditional Statements: A If and Only If B
If x = 3, then x + 1 = 4, and if x + 1 = 4, then x = 3. Symbolically, we would write this as (A → B) Λ (B → A).
We even have a shorter form of this:
(A → B) Λ (B → A) may be written as A ( B.
The statement A ( B is known also as "A if and only if B". where A ( B is equivalent to (A → B) Λ (B → A)
Example: Let F = "My car is fast", N = "My car is new", and R="My car needs repair". Represent each of the statements below in symbolic form:
A) My car is slow and needs to be repaired.
B) It is not the case that if my car is fast, then it needs to be repaired.
C) It is not the case that my car is old if and only if it is not fast.
Solutions:
A) (~F) Λ R Note that extra parentheses eliminate any confusion with ~F Λ R
B) ~(F → R)
C) ~(~N ( ~F)
Final Exam Question: Be able to convert words to symbols as was done in the previous example.
Example: Let F = "My car is fast", N = "My car is new", and R="My car needs repair". Represent each of the statements below in word form:
A) F → ~R
B) (F Λ ~N) V R
C) F ( ~R
Solutions:
A) If my car is fast, then it does not need repair.
B) Either it is the case that my car is both fast and is not new or it is the case that it needs repair.
C) My car is fast if and only if it does not need repair.
Note that in example B) we use the words "it is the case" to signify the parentheses around F Λ~N.
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Truth Tables for Negation, Conjunction, and Disjunction
Truth Values of Conjunctions and Disjunctions
Conjunctions are only true when both statements are true.
If M represents the statement “The moon is green with yellow spots” and Y represents the statement “The sun is hot”, the statement "The moon is green with yellow spots and the sun is hot" may be written M Λ Y. Since "The moon is green with yellow spots" is always false, this statement M Λ Y is false.
Disjunctions are true if either of the statements is true.
If M represents the statement “The moon is green with yellow spots” and Y represents the statement “The sun is hot”, the statement "The moon is green with yellow spots or the sun is hot" may be written M V Y. Since "The sun is hot" is always true, this statement M V Y is true, even though one component was false.
In the previous two examples, the components of the conjunction and disjunction were known to be true or false. For example, we know that "the sun is hot" is always a true statement. In general, we must account for all possible truth values. We may construct what is called a “truth” table to summarize all possible truth values.
Example: We may summarize all possibilities for the truth value of the statement P Λ Q with the truth table shown below.
|P |Q |P Λ Q |
|T |T |T |
|T |F |F |
|F |T |F |
|F |F |F |
Note that every combination of truth values is given in the left two columns and the resulting truth value is given in the right-most column. P Λ Q is only true when both P and Q are true.
Example: We may summarize all possibilities for the truth value of the statement P V Q with the truth table shown below.
|P |Q |P V Q |
|T |T |T |
|T |F |T |
|F |T |T |
|F |F |F |
Note that every combination of truth values is given in the left two columns and the resulting truth value is given in the right-most column. P V Q is true when either P or Q are true.
Truth Values of Conjunctions and Disjunctions Combined With Negations
Example: We may summarize all possibilities for the truth value of the statement ~P V ~Q with the truth table shown below.
|P |Q |~P |~Q |~P V ~Q |
|T |T |F |F |F |
|T |F |F |T |T |
|F |T |T |F |T |
|F |F |T |T |T |
Note that we must construct additional columns for ~P and ~Q. Then again, every combination of truth values is given in the left two columns and the resulting truth values are given in the 3 right-most columns. The truth value of the negation of a statement is always the opposite as the truth value of the un-negated statement. ~P V ~Q is true when either ~P or ~Q are true.
Constructing Truth Tables
To construct a truth table containing two variables, use the following procedure:
1. There are two columns with 4 rows underneath with the 4 possibilities of True and False. These two columns are at the far left.
2. There are additional columns for negations of variables if needed. The negations will have the opposite truth values.
3. There are additional columns for each major component of the final statement.
4. The last column at the right contains the final statement.
Example: Construct a truth table for the statement (P V ~Q) Λ ~P
1. Construct a table that has two columns for P & Q with the 4 possibilities for T & F.
|P |Q | | | | |
|T |T | | | | |
|T |F | | | | |
|F |T | | | | |
|F |F | | | | |
2. Make columns for ~P and ~Q
|P |Q |~P |~Q | | |
|T |T |F |F | | |
|T |F |F |T | | |
|F |T |T |F | | |
|F |F |T |T | | |
Steps 3 and 4 on NEXT PAGE!
3. Make a column for P V ~Q
|P |Q |~P |~Q |P V ~Q | |
|T |T |F |F |T | |
|T |F |F |T |T | |
|F |T |T |F |F | |
|F |F |T |T |T | |
4. Make a column for (P V ~Q) Λ ~P
|P |Q |~P |~Q |P V ~Q |(P V ~Q) Λ ~P |
|T |T |F |F |T |F |
|T |F |F |T |T |F |
|F |T |T |F |F |F |
|F |F |T |T |T |T |
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Example: Construct a truth table for (~Q) V(P^Q)
Final Exam Question: Construct a Truth Table for a statement that combines P and Q with negation ~ and connectives ^ and V.
Unit 3 - Truth Tables for Conditional & Biconditional and Equivalent Statements & DeMorgan's Laws
Conditional Statements (If-Then Statements)
The truth table for P → Q is shown below.
|P |Q |P → Q |
|T |T |T |
|T |F |F |
|F |T |T |
|F |F |T |
Here's all you have to remember:
If-Then statements are ONLY false when the IF-PART is TRUE and the THEN-PART is false. If the IF-PART is not true, then the THEN-PART does not have to follow.
The Converse
The "Converse" of P→ Q is Q → P. The truth table for Q → P compared to P → Q is shown below.
|P |Q |Q → P |P → Q |
|T |T |T |T |
|T |F |T |F |
|F |T |F |T |
|F |F |T |T |
Note: The Converse is NOT logically equivalent to the original conditional. We can never switch the If-Part with the Then-Part in a conditional to give us an equivalent statement for all cases.
The Inverse
The "Inverse" of P→ Q is ~P→ ~Q. The truth table for ~P → ~Q compared to P → Q is shown below.
|P |Q |~P |~Q |~P → ~Q |P → Q |
|T |T |F |F |T |T |
|T |F |F |T |T |F |
|F |T |T |F |F |T |
|F |F |T |T |T |T |
Note: The Inverse is NOT logically equivalent to the original conditional. We can never take the negations of both the If-Part and the Then-Part in a conditional to give us an equivalent statement for all cases. Note however that the converse is logically equivalent to the inverse.
The Contrapositive
The "Contrapositive" of P→Q is ~Q→~P. The truth table for ~Q → ~P compared to P→Q is shown below.
|P |Q |~P |~Q |~Q → ~P |P → Q |
|T |T |F |F |T |T |
|T |F |F |T |F |F |
|F |T |T |F |T |T |
|F |F |T |T |T |T |
Note: The Contrapositve IS logically equivalent to the original conditional. We can always take the negations of both the If-Part and the Then-Part in a conditional and switch them with each other to give us an equivalent statement for all cases. This is the reason that the converse is logically equivalent to the inverse - the if and then-parts are switched and negations are taken.
Example: Write the converse, inverse, and contrapositive of the statement: "If it rains on me, then I get wet."
Converse: "If I get wet, then it has rained on me".
Inverse: "If it does not rain on me, then I don't get wet".
Contrapositive: If I do not get wet, then it did not rain on me.
Biconditional Statements (If-and-only-If Statements)
P if and only if Q is represented as P ( Q. The truth table for P ( Q is shown below.
|P |Q |P ( Q |
|T |T |T |
|T |F |F |
|F |T |F |
|F |F |T |
Here's all you have to remember: If-and-only-if statements are ONLY true when P and Q are BOTH TRUE or when P and Q are BOTH FALSE.
Final Exam Question: Know how to do a truth table for variations of P -> Q and P ( Q and also know what P ( Q means.
Alternative Truth Table For P ( Q
Since P ( Q is logically equivalent to (P -> Q)Λ(Q -> P). The truth table for P ( Q written in this way is shown below.
|P |Q |P → Q |Q → P |(P → Q) Λ (Q → P) |
|T |T |T |T |T |
|T |F |F |T |F |
|F |T |T |F |F |
|F |F |T |T |T |
Here's another way to remember if-and-only-if statements:
P If-and-only-if Q means that events P and Q either both happen or both don't happen.
Example: x = 2 if and only if x+3 = 5 would only be false if one statement were true and the other false - something that is impossible in mathematics as we know it. For all practical purposes, only 2 of the 4 rows of the if and only if table shown above are possible for this mathematics example. So we conclude that x = 2 if and only if x+3 = 5 is a TRUE statement.
Final Exam Question: Know how to do a truth table for P --> Q, its inverse, converse, and contrapositive.
In Summary:
An If-then statement is only false if the IF-Part is true yet the THEN-Part is false. In the Venn diagram below, this is shown by placing an X within the A circle (IF-Part True) that lies outside the B circle (THEN-Part False). The result would be a false If A, Then B statement. If, however, the X was within the
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If the IF-part is False, the IF-THEN statement is still true. Think of this as a scientific theory that you are trying to disprove. The IF-Part is the hypotheses of the theory. To disprove the theory, you must first satisfy the hypothesis and then find a case where the conclusion (the Then-part), does not follow. For example, if someone theorized, "If a 12-gauge wire is made of plastic, then it can be used to transmit electricity as well as a copper wire", you would disprove this theory by constructing a 12-gauge wire made of plastic and then showing it does not conduct electricity as well as a copper wire of the same gauge. You could not disprove the theory by testing a wire made of aluminum! The figure below illustrates this for IF A, THEN B.
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A IF-AND-ONLY-IF B really means A and B represent the same events or events that occur (or don't occur) simultaneously. Using the Venn Diagram, this would mean A and B represent the same circle as shown below. A If-and-only-if B statements are true if BOTH A & B parts are true or BOTH A & B parts are false.
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Example: Is the statement "A car will not start if and only if the car is out of gas" an example of an if-and-only-if statement that is always true in a real-life situation?
ANSWER: No. The car could have plenty of gas but have a dead battery. This would make one part of the statement true while the other part is false. Both parts P and Q of PQ must be true at the same time or both parts must be false at the same time.
De Morgan's Laws
If you construct a truth table for ~(P Λ Q) and a truth table for ~P V ~Q you can see that the resulting truth values on the far right are exactly the same for the same set of truth values for P & Q.
|P |Q |~P |~Q |~P V ~Q |
|T |T |F |F |F |
|T |F |F |T |T |
|F |T |T |F |T |
|F |F |T |T |T |
|P |Q |PΛ Q |~(P Λ Q) |
|T |T |T |F |
|T |F |F |T |
|F |T |F |T |
|F |F |F |T |
We say that ~P V ~Q is logically equivalent to ~(P Λ Q) because both final columns on the far right have the same truth values.
Whenever two statements have the same truth values in the far right column for the same starting values of the variables within the statement we say the statements are logically equivalent.
De Morgan's Laws are two special cases of logical equivalence. De Morgan's Laws state:
~P V ~Q is logically equivalent to ~(P Λ Q)
~P Λ ~Q is logically equivalent to ~(P V Q)
You may show these equivalent relationships with truth tables.
Example: Find the negation of (P V ~Q) and also the negation of (~P ^ ~Q).
Solution: DeMorgan's Laws basically say that if you negate an OR or AND statement, you negate each component and switch the OR to AND or switch AND to OR.
~(P V ~Q) ≡ ~P ^ Q
~(~P ^ ~Q) ≡ P V Q
Example: Rewrite "I will stay or I will go" as an equivalent AND statement by taking a negation, applying DeMorgan's Laws, and then taking another negation.
If we let S = "I will stay" and G = "I will go", then our original statement is S V G.
A single negation, results in ~(S V G).
Applying DeMorgan's Laws allows us to write this as ~S ^ ~G.
Now, taking the negation of this results in ~ ( ~S ^ ~G).
In words, this is It is not the case that I will not stay and also not go.
Final Exam Question: Know how to apply DeMorgan's Law in order to rewrite the negation of an AND statement or the negation of an OR statement.
Unit 4 - Arguments & Truth Tables and Euler Diagrams Applied to False Advertising
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Fallacy of the Converse
Here, people assume that because P→Q is true, then Q→P is also true. This is known as the Fallacy of the Converse.
More formally: Assuming that P→Q is true and we are given that event Q occurs, we falsely conclude that event P follows.
Example: Assume it true that "If you have the flu, then you will have a fever". Also assume it is true that a doctor's patient named Bob has a fever. Can we conclude that Bob has the flu?
The answer here is no. If we did conclude that Bob has the flu based only on the fact that he has a fever, we would be illustrating the Fallacy of the Converse. A person could have a fever from an infection such as strep throat or many other causes. In this particular case, the fever indicates that Bob very likely may have the flu since most people with a fever have the flu, but the fever alone does not indicated it with 100% accuracy. We would have to look at other symptoms and make sure that there is not some type of infection or other condition causing the flu.
We may show that this is a fallacy with a truth table by showing that [(P→Q) ^ Q]→P does not result in ALL TRUE values, as shown below.
|P |Q |P→Q |(P→Q) ^ Q |[(P→Q) ^ Q]→P |
|T |T |T |T |T |
|T |F |F |F |T |
|F |T |T |T |F |
|F |F |T |F |T |
What we are doing here is showing that we are accepting the statement P→Q as our premise and we are also accepting Q as one of our premises. We then show that the conclusion P does not follow with a true value in all cases. The one FALSE value in the far right column shows that there is a case where assuming that (P→Q) and Q to both be true does not result in P following as a true statement. In other words, that one false value in the 3rd row, far right, shows that Q→P does not always follow from P→Q.
Symbolic Forms of Arguments
To write an argument in symbolic form, we list the premises and then below the premises we list the conclusions. An example of Direct Reasoning is given below:
PREMISE 1: "If I have a fever, then I am sick"
PREMISE 2: "I have a fever"
-------------------------------------------------------
CONCLUSION: "I am sick"
The Fallacy of the Converse in Symbolic Form
Here is the previous example of the Fallacy of the Converse, in symbolic form.
PREMISE 1: "If a person has the flu, then this person has a fever" (assumed to be true)
PREMISE 2: "Bob has a fever"
-------------------------------------------------------
CONCLUSION: "Bob has the flu" ................
................
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