# discrete mathematics notes for first year students, Exercises for Discrete Mathematics

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R. Johnsonbaugh, Discrete Mathematics 5th edition, 2001

TR1333: DISCRETE MATHEMATICS

: I I

LECTURE 5: MATHEMATICAL SYSTEM

What are the two important questions that arise in the study of

mathematics ?? 1. When is a mathematic al argument

correct? 2. What method can be used to

construct mathematical arguments?

Mathematical System

• A mathematical system consists of – Undefined terms – Definitions – Axioms

Undefined Terms Undefined terms are

the basic building blocks of a mathematical system.

• These are words that

are accepted as starting concepts of a mathematical system.

Example 15: In Euclidean geometry we have undefined terms such as:

- Point - Line

Definitions

• A definition is a proposition constructed from undefined terms and previously accepted concepts in order to create a new concept.

Example 16. In Euclidean geometry the following are definitions:

•Two triangles are congruent if their vertices can be paired so that the corresponding sides are equal and so are the corresponding angles.

•Two angles are supplementary if the sum of their measures is 180 degrees.

Axioms • An axiom is a

proposition accepted as true without proof within the mathematical system.

• There are many examples of axioms in mathematics.

Example 17: In Euclidean geometry the following are axioms:

•Given two distinct points, there is exactly one line that contains them.

•Given a line and a point not on the line, there is exactly one line through the point which is parallel to the line.

Theorems • A theorem is a

proposition of the form p  q which must be shown to be true by a sequence of logical steps that assume that p is true, and use definitions, axioms and previously proven theorems.

Example 18: •If two sides of a triangle is equal, then the angle opposite them are equal.

•If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram

Lemmas And Corollaries

• A lemma is a small theorem which is used to prove a bigger theorem.

• A corollary is a theorem that can be proven to be a logical consequence of another theorem. – Example from Euclidean geometry: "If the three sides of a triangle have equal

length, then its angles also have equal measure."

Types of Proof

• A proof is a logical argument that consists of a series of steps using propositions in such a way that the truth of the theorem is established.

• Types of Proof - Direct proof - Indirect proof

Direct Proof Direct proof: p  q

• A direct method of attack that assumes the truth of proposition p, axioms and proven theorems so that the truth of proposition q is obtained.

Direct Proof Example 19:

For a real numbers d, d1, d2 and x ifd = min {d1, d2} and x d, thenx d1 and x d2Proof - assume that d, d1, d2 and x are arbitrary real numbers

-it suffices to assume that,

d = min {d1, d2} and x d is true, and then prove that

x d1 and x d2 is true

Direct Proof • Proof (continue):

from the definition of min, it follows that d d1 and d d2

from x d and d d1, we may derive x d1 (theorem: for all real numbers x, y and z,

if xy and yz, then xz) from x d and d d2, we may derive x d2

(from the same above theorem) therefore x d1 and x d2

Indirect Proof The method of proof by contradiction of a theorem p  q consists of the following steps:

1. Assume p is true and q is false 2. Show that ~p is also true. 3. Then we have that p ^ (~p) is true. 4. But this is impossible, since the statement p^ (~p) is always false. There is a

contradiction! 5. So, q cannot be false and therefore it is true.

Indirect Proof Example 20: For a real numbers x and y,

if x + y ≥ 2, then either x ≥ 1 or y ≥ 1 Proof:

- suppose that the conclusion is false - then x < 1 and y < 1 - we may add inequalities to obtain

x + y < 1 + 1 = 2 - at this point, we have derived the

contradiction p ^ ~p where p : x + y ≥ 2

- thus we conclude that the statement is true.

Indirect Proof

OR: show that the contrapositive (~q)(~p) is true. Since (~q)  (~p) is logically equivalent to p 

q, then the theorem is proved.

Indirect Proof Example 21: For a real numbers x and y, if x + y ≥ 2, then either x ≥ 1 or y ≥ 1

- Let p : x + y ≥ 2 and q : x ≥ 1 or y ≥ 1 - p → q (if x + y ≥ 2, then either x ≥ 1 or y ≥ 1)

- since (~q)  (~p) is logically equivalent to p  q, show that the contrapositive

(~q)(~p) is true (use truth table)

Indirect Proof p q p → q (~q) → (~p)

T T T T F F F T T F F T