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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
if**d *= min {*d1*, *d2*} and *x *≤ *d*, *then**x *≤ *d1* and *x
*≤ *d2***Proof
**- 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 *x* ≤ *y* and *y* ≤ *z*, then *x* ≤ *z*)
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