Math 237- Mathematical Thinking

What is Mathematical Thinking?

Mathematical thinking is the use of logic, definitions, patterns, relationships, and proof to understand mathematical ideas. In discrete mathematics, the goal is often not simply to calculate an answer, but to explain why a statement is true or false.

A useful approach: Understand the definitions → identify what is known → choose a strategy → work logically → justify each step → check the conclusion.

1. Logical Reasoning

Logic provides a language for expressing mathematical statements and determining whether conclusions follow from premises.

Statements

A statement is a sentence that is either true or false, but not both.

For example, let:

p : 7 is prime

This is a statement and its true.

Logical Connectives

Symbol	Name	Meaning
$\neg p$	Negation	Not $p$
$p \land q$	Conjunction	$p$ and $q$
$p \lor q$	Disjunction	$p$ or $q$
$p \to q$	Implication	If $p$ , then $q$
$p \leftrightarrow q$	Biconditional	$p$ if and only if $q$

Example: Implication

Consider:

p \to q

p

is true and

q

is false, the implication is false. In every other truth-value combination, the implication is true.

Practice

Problem 1: Let

p

mean “8 is even” and

q

mean “8 is prime.” Write

p \land q

p \land q

is False since * is not prime.

2. Proof Techniques

A mathematical proof is a logical argument showing that a statement follows from definitions, known facts, and previously established results.

Direct Proof

To prove an implication

p \to q

, assume

p

is true and show that

q

follows.

Example: Prove that the sum of two even integers is even.

Step 1: Let the even integers be

2 m

and

2 n

Step 2: Add them up.

2 m + 2 n = 2 (m + n)

Step 3: Since

m + n

is an integer,

2 (m + n)

is even.

Conclusion: The sum of two even integers is even.

Proof by Contrapositive

Instead of proving

p \to q

, prove its logically equivalent contrapositive:

\neg q \to \neg p

Proof by Contradiction

Assume the statement to be proved is false and show that this assumption leads to a contradiction.

Proof by Cases

Divide the problem into cases that cover all possible situations, then prove the statement in each case.

Mathematical Induction

Induction is commonly used to prove statements about positive integers.

1. Base case: Show the statement is true for the first value.

2. Inductive hypothesis: Assume it is true for

n = k

3. Inductive step: Show it is then true for

n = k + 1

Example: Show the first

n

positive integers have sum

1 + 2 + \dots + n = \frac{n (n + 1)}{2}

The base case

n = 1

gives

1 = 1

. The inductive step assumes the formula for

k

, adds

k + 1

, and simplifies to the formula for

k + 1

Practice: Which proof method would be especially appropriate for a statement involving all positive integers and a formula containing

n

Mathematical induction.

3.Sets

A set is a well-defined collection of distinct objects called elements.

A = {1,2,3,4}

Basic Set Operations

Operation	Symbol	Meaning
Union	$A \cup B$	Elements in $A$ , $B$ , or both
Intersection	$A \cap B$	Elements common to both
Difference	$A - B$	Elements in $A$ but not $B$
Complement	$A^{c}$	Elements outside $A$ in the universal set
Subset	$A \subseteq B$	Every element of $A$ is in $B$

Let

A = {1,2,3,4}

and

B = {3,4,5,6}

Let

A = {1,2,3,4}

and

B = {3,4,5,6}

A \cup B = {1,2,3,4,5,6}

A - B = {1,2}

Practice: Let

A = {2,4,6,8}

and

B = {1,2,3,4}

. Find

A \cap B

{2,4}

Math 237- Mathematical Thinking

What is Mathematical Thinking?

1. Logical Reasoning

Statements

Logical Connectives

Example: Implication

Practice

Answer+

2. Proof Techniques

Direct Proof

Proof by Contrapositive

Proof by Contradiction

Proof by Cases

Mathematical Induction

Answer+

3.Sets

Basic Set Operations

Answer+