FTCE Mathematics 6-12: Basics of Mathematical Reasoning

Premise and Argument
A premise is a statement that precedes a conclusion, in an argument. It is the proposition, or assumption, of an argument. An argument will have two or more premises.
Example:
If it is hot, then I will go swimming. (Premise)
It is hot. (Premise)
Therefore, I will go swimming. (Conclusion)

Simple and Compound Statements
A statement in propositional logic is any sentence or expression that has a truth value—that is, that may in principle be considered true or false. 'Hello!', for example, is not a statement; there's no sense in which it could be considered true or false. On the other hand, '
'
is a statement, albeit a clearly false one. A compound statement is a statement that contains one or more other statements, combined or modified in some way. A simple statement, or a proposition, is a statement that cannot be broken down further into smaller statements.
For example, consider the statement, 'If it rains tomorrow, then the streets will be flooded and traffic will be slow.' This is a compound statement, because two smaller statements are embedded within it: 'it rains tomorrow' and 'the streets will be flooded and traffic will be slow.' the former cannot be broken down further, and is therefore a simple statement, but the latter is another compound statement, because it includes two simple statements: 'the streets will be flooded' and 'traffic will be slow.'
Common operations used to combine and modify simple statements into compound statements include conjunction, disjunction, negation, and implication.

Conjunction
Conjunction is an operation that combines two or more statements into a compound statement that is true if and only if all of the component statements are true. When the statements are written out, conjunction is marked by the word 'and.' For example, the statement 'Roses are red and violets are blue' is a compound statement that is a conjunction of the two simple statements 'roses are red' and 'violets are blue.'
Conjunction is represented by the operator
.
In the previous example, if proposition P is 'roses are red' and proposition Q is 'violets are blue,' then the compound statement 'roses are red and violets are blue' would be written as '
.'

Disjunction
Disjunction is an operation that combines two or more statements into a compound statement that is true if and only if at least one of the component statements is true. When the statements are written out, conjunction is marked by the word 'or.' For example, the statement 'I'll clean the closet today, or you'll clean it tomorrow' is a compound statement that is a disjunction of the simple statements 'I'll clean the closet today' and 'you'll clean the closet tomorrow.'
Disjunction is represented by the operator
.
In the previous example, if proposition P is 'I'll clean the closet today' and proposition Q is 'you'll clean the closet tomorrow,' then the compound statement 'I'll clean the closet today, or you'll clean it tomorrow' would be written as '
.'
Note that while in everyday language 'or' implies exclusivity, as one or the other option but not both, in formal logic disjunction includes the possibility of both propositions being true. If I clean the closet today and you clean it tomorrow, then the sample statement 'I'll clean the closet today, or you'll clean it tomorrow' is true.

Conditional and Biconditional Statements
A conditional statement, or implication, is a compound statement of the form 'if P, then Q.' for instance, the statement 'if a snake has rattles, then it's venomous' is a conditional statement. It can also be written 'P implies Q' ('that a snake has rattles implies that it's venomous') or 'P only if Q' ('a snake has rattles only if it's venomous'). the conditional statement is true when both P and Q are true, and even when P is false, regardless of Q. In other words, the conditional statement is false only when P is true and Q is false. the conditional statement is written as '
is called the premise of the statement, and Q the conclusion.
A biconditional statement is a combination of two conditional statements: both P, then Q' and 'if Q, then P.' To state it more concisely, 'P if and only if Q.' an example is 'A number is even if and only if it is divisible by 2.' the biconditional statement is true when P and Q are either both true or both false; it is false if P is true and Q is false or vice versa. the biconditional statement is written '
.'

Existential and Universal Quantifiers
A quantifier is a logical construct used in a compound statement to give information on the number of subjects for which a statement is true. The existential quantifier specifies that the statement is true for at least one subject. It is usually introduced as 'there exists,' such as 'there exists a real number that is equal to its own square.' The symbol
universal quantifier specifies that the statement is true for every subject.
It is usually said as " /> is used to signify the universal quantifier. For example, the preceding statement could be written as

Sometimes it is useful to specify that there exists exactly one subject that meets a given criterion. In other words, there exists a unique such subject. This can be signified by adding an exclamation mark after the existential quantifier: '
' means 'there exists a unique real number that has an absolute value of zero.'

Truth Tables
A truth table shows the truth value of one or more compound statements for each possible combination of truth values of the propositions within it. The truth table contains one column for each proposition, and one column for each of the compound statements to be analyzed. It has one row for each combination of truth values of the propositions. Since each proposition has two possible truth values, the number of possible combinations for N propositions is
, so the truth table will have
 rows.
The table below is a simple truth table for some common compound statements involving two propositions.




Truth tables are useful for comparing two compound statements to see if they are equivalent. They are also useful for analyzing whether complicated compound statements are true or false. A table can have a column for each of the increasingly complex compound statements that combine to form the total statement.

Truth Table to Validate the Rule of Detachment
The Rule of Detachment states that given the premises,
, the valid conclusion is Q.
In other words, for every case where


Notice the first cell under
was also true.

Truth Table to Validate the Chain Rule
The Chain Rule states that given the premises,
, the valid conclusion is
. case where
 is true,
 will also be true. The trith table below illustrates this fact:




Notice that for every case where
 was true,
 was also true.
For example, consider the premises below:
If I hike a mountain, I will not eat a sandwich.
If I do not eat a sandwich, I will drink some water.
I will not drink some water.
Write a valid conclusive statement. Explain how you arrived at your answer. Be specific in your explanation.
Valid conclusive statement: I will not hike a mountain.
Application of the chain rule and rule of contraposition give the valid conclusion of
. According to the chain rule, given
 and
, then
. According to the rule of contraposition,
 and
 yields
. On a truth table, for every place where
 is true,
 is also true. Thus, this is a valid conclusive statement.

Negation
In propositional logic, negation refers to the inversion of the truth value of a statement. The negation of a statement is true if the original statement is false, and false if the original statement is true.
Negation can be signified by the word 'not.' For instance, the negation of the statement 'Bob's cat is black' is 'Bob's cat is not black.' the symbol for negation is
. So, if the proposition P is 'Bob's cat is black,' then
 would be 'Bob's cat is not black.' Two negations cancel each other out:
, or 'not not P equals P.'
It's important to be careful when combining negation with quantifiers.
The negation of 'there exists a cow that has a blue horn' is not 'there exists a cow that does not have a blue horn.' Rather, the negation is 'there does not exist a cow that has a blue horn.' the statements
 and
 are not equivalent. the same is true for the universal quantifier:
 and
 do not mean the same thing. There is, however, a connection between the negated quantifiers:
 is equivalent to
, and
 is equivalent to
.

De Morgan's Laws

De Morgan's Laws are a set of useful relations connecting negation, conjunction, and disjunction. They can be stated briefly as 'the negation of a conjunction is the disjunction of the negations' and 'the negation of a disjunction is the conjunction of the negations.' This can be written as:

For instance, if proposition P is '
' and proposition Q is '
', then De Morgan's
Laws state that 'it is not true that both
are greater than 3" /> is not greater than 3, or y is not greater than 3.'
The validity of De Morgan's Laws can be shown with truth tables. Note that the columns highlighted in the same color match:




Converse, Inverse, and Contrapositive of a Conditional Statement
The converse of a conditional statement is a conditional statement with the premise and conclusion interchanged. the converse of
 is
inverse has the premise and conclusion both negated. The inverse of
contrapositive has the premise and conclusion both negated and interchanged. The contrapositive of &  is
.
For example, given the conditional statement, 'If there is a key in the lock, then someone is home,' the converse is 'if someone is home, then there is a key in the lock,' the inverse is 'if there is not a key in the lock, then no one is home,' and the contrapositive is 'if no one is home, then there is not a key in the lock.'
Note that the converse and inverse of a statement are not logically equivalent to the original statement. For instance, the statement 'if
, then
' is true, but its converse 'if
, then
' is not, because there are many numbers greater than 2, not just 4. A statement is, however, logically equivalent to its contrapositive—and the converse and inverse, while not equivalent to the original statement, are equivalent to each other.
The following truth table summarizes the relationships:



Inductive Reasoning
Inductive reasoning is a method used to make a conjecture, based on patterns and observations. The conclusion of an inductive argument may be true or false.

Mathematical Example:
A cube has 6 faces, 8 vertices, and 12 edges. A square pyramid has 5 faces, 5 vertices, and 8 edges. A triangular prism has 5 faces, 6 vertices, and 9 edges. Thus, the sum of the numbers of faces and vertices, minus the number of edges, will always equal 2, for any solid.

Non-Mathematical Example:
Almost all summer days in Tucson are hot. It is a summer day in Tucson. Therefore, it will probably be hot.

Deductive Reasoning
Deductive reasoning is a method that proves a hypothesis or set of premises. The conclusion of a valid deductive argument will be true, given that the premises are true. Deductive reasoning utilizes logic to determine a conclusion. For instance, consider the following application of the chain rule:
If a ding is a dong, then a ping is a pong.



If a ping is a pong, then a ring is a ting.


A ding is a dong.



Therefore, a ring is a ting.



Formal Reasoning
Formal reasoning, in mathematics, involves justification using formal steps and processes to arrive at a conclusion. Formal reasoning is utilized when writing proofs and using logic. For example, when applying logic, validity of a conclusion is determined by truth tables. A set of premises will yield a given conclusion. This type of thinking is formal reasoning. Writing a geometric proof also employs formal reasoning. For example:
If a quadrilateral has four congruent sides, it is a rhombus.
If a shape is a rhombus, then the diagonals are perpendicular.
A quadrilateral has four congruent sides.
Therefore, the diagonals are perpendicular.

Informal Reasoning
Informal reasoning, in mathematics, uses patterns and observations to make conjectures. The conjecture may be true or false. Several, or even many, examples may show a certain pattern, shedding light on a possible conclusion. However, informal reasoning does not provide a justifiable conclusion. A conjecture may certainly be deemed as likely or probable.
However, informal reasoning will not reveal a certain conclusion. Consider the Mathematical Idea – Given a sequence that starts with 1 and each term decreases by a factor of
, the limit of the sum of the sequence will be 2.
Informal Reasoning – The sum of 1 and
 is
. The sum of 1,
, and
 is
. The sum of 1,
,
, and
 is
, Thus, it appears that as the sequence approaches infinity, the sum of the sequence approaches 2.

Proofs
A proof serves to show the deductive or inductive process that relates the steps leading from a hypothesis to a conclusion. A proof may be direct (
, meaning that a conclusion is shown to be true, given a hypothesis. There are also proofs by contradiction (
, whereby the hypothesis is assumed to be true, and the negation of the conclusion is assumed to be true. (In other words, the statement is assumed to be false.) Proofs by contraposition
show that the negation of the conclusion leads to the negation of the hypothesis. (In other words, the negation of the conclusion is assumed to be true, and it must be shown that the negation of the hypothesis is also true.) A mathematical induction proof seeks to show that
 is true and that
 is true, given that
 is true. Direct proofs, proofs by contradiction, and proofs by contraposition use deductive methods, while a mathematical induction proof uses an inductive method.
Direct proofs are those that assume a statement to be true. The purpose of such a proof is to show that the conclusion is true, given that the hypothesis is true. A sample of a direct proof is shown below:
Prove 'If m divides a and m divides b, then m divides
.'

Proof:
Assume m divides a and m divides b.
Thus, a equals the product of m and some integer factor, p, by the definition of division, and b equals the product of m and some integer factor, q, by the definition of division. According to substitution,
 may be rewritten as
. Factoring out the m gives
divides
 is an integer, according to the closure property, we have shown that m divides
, by the definition of division.
Indirect proofs (or proofs by contradiction) are those that assume a statement to be false. The purpose of such a proof is to show that a hypothesis is false, given the negation of the conclusion, indicating that the conclusion must be true. A sample of an indirect proof is shown below:
Prove 'If
 is odd, then
 is even.'

Proof:
Assume
 is odd and
 is odd.
· According to the definition of odd,
, where a is an element of the integers.
Thus, by substitution,
, which simplifies as
, or
, which may be rewritten as
. Any even integer may be written as the product of 2 and some integer, k. Thus, we have shown the hypothesis to be false, meaning that the conditional statement must be true.
A proof by contraposition is one written in the form,
. In other words, a proof by contraposition seeks to show that the negation of Q will yield the negation of P. A sample of a proof by contraposition is shown below:
Prove 'If
 is even, then
 is odd.'
Assume that if
 is even, then
 is
Assume
 is even.
Thus, by the definition of an even integer,
. B. substitution,
 may be rewritten as
, which simplifies as
. This expression cannot be written as the product of 2 and some factor, k. Thus,
 is odd, by definition of an odd integer. So, when
 is even,
 is odd, according to contraposition.
A proof by contradiction is one written in the form,
.

In other words, a proof by contradiction seeks to show the negation of q will result in a false hypothesis, indicating that the conclusion of the statement, as written, must be true. In other words, the conditional statement of
 is true.

Mathematical Induction Proof Utilizing Inductive Reasoning
A mathematical induction proof utilizes inductive reasoning in its assumption that if
 is true, then
 is also true. The induction hypothesis is
. This step utilizes inductive reasoning because an observation is used to make the conjecture that
 is also true.
For all natural numbers, n, the sum is equal to
.

Show that
 is true.
.
Assume P(k) is true.
.
This previous step is the inductive hypothesis. This hypothesis may be used to write the conjecture that
 is also true:












Problems:

P1. Given the following statements:
If I get a bonus, then I will go on vacation.
If I go on vacation, then I will visit Egypt.
I got a bonus.
Therefore, I will visit Egypt.
Rewrite the statements in logical notation.
Make a truth table for the scenario.
Determine if the argument is logical.
(d) TRUE or FALSE: It would contradict this argument to visit Egypt and not have received a bonus.
P2. Given the: I will run a marathon in
January if and only if the Cubs win the World Series.
I will not run a marathon in
January.
Therefore, the Cubs did not win the World Series.
(a) Rewrite the statements in logical notation.
(b) Make a truth table for the scenario.
(c) Determine if the argument is logical.
(d) TRUE or FALSE: It is logically equivalent to the first premise above to say, 'I will not run a marathon in January if and only if the Cubs do not win the World Series.
P3. Use proof by induction to demonstrate: 'The sum of n odd natural numbers is equal to
.'
P4. Use informal reasoning to justify the statement, 'If n is a whole number, then
 is odd.' Explain the reasoning steps used.
P5. Use formal reasoning to justify the statement, 'If a divides b, a divides c, and a divides d, then a divides the sum of b, c, and d.' Show the formal proof.

Solutions:
P1. (a) Let us say
,
, and
:
If I get a bonus, then I will go on vacation.


If I go on vacation, then I will visit Egypt.

I got a bonus.


Therefore, I will visit Egypt.


(b) The truth table for the scenario:




(c) This is an instance of the chain rule and is logically sound.
(d) FALSE: It would NOT contradict this argument to visit Egypt and not have received a bonus. Look at the rows of the truth table where
 and
, in these cases both statements hold:



P2. (a) Let us say
 and
:
I will run a marathon in January if and only if the Cubs win the World Series.

I will not run a marathon in January.


Therefore, the Cubs did not win the World Series.


(b) The truth table for the scenario:



(c) Based on the truth table, it is clear that
 is logically equivalent to
.
(d) TRUE: The following truth table demonstrates the equivalence:





P3. Show that
 is
 .
Assume
 is
.
We want to show that
.
.
We can distribute 2 to each of the k terms:
.
.
Rearrange and simplify:
.
 is true.
Thus, according to mathematical induction,
.

P4. Using the sequence, 0, 1, 2, 3, 4, 5, 6, … for n, evaluation of the expression,
, gives
,
,
,
,
,
, and
, or 1, 3, 7, 13, 21, 31, and 43, all of which are odd numbers. Thus, it appears that given any whole number, n, evaluation of the expression
 will yield an odd number.

P5. Direct Proof:
Assume a divides b, a divides c, and a divides d.
Given the definition of divides, a divides b indicates that there exists some integer, r, such that
divides c indicates that there exists some integer, s, such that
gives
 is an integer, according to the closure property under addition. Thus, a divides the sum of b, c, and d.