1 Introduction to Abstract Mathematics The Logic and Language of Proofs Instructor: Hayk Melikya [email protected] Proposition: Informal introduction to

1Introduction to Abstract Mathematics

The Logic and Language of Proofs

Instructor: Hayk Melikya [email protected]

Proposition: Informal introduction to logic.

Definition: A proposition ( or statement) is a sentence that is either TRUE or FALSE. He is handsome. WHAT TIME IS IT?. 5 + 7. Please open the door.


Logic

Logic is not only the foundation of mathematics, butalso is important in numerous fields including law,medicine, and science. Although the study of logicoriginated in antiquity, it was rebuilt and formalized inthe 19th and early 20th century. George Boole (Booleanalgebra) introduced mathematical methods to logic in1847 while Georg Cantor did theoretical work on setsand discovered that there are many different sizes ofinfinite sets.


Statements or Propositions A proposition or statement is a declaration which is

either true or false. Some examples: 2+2 = 5 is a statement because it is a false

declaration. Orange juice contains vitamin C is a statement that is

true. Open the door. This is not considered a statement

since we cannot assign a true or false value to this sentence. It is a command, but not a statement or proposition.


Negation

The negation of a statement, p , is “not p” and is denoted by ┐ p

Truth table: p ┐ p T F F T If p is true, then its negation is false. If p is false, then its

negation is true.


Conjuction In ordinary English, we are building new propositions from

old ones via connectors. Similar way we will construct new

propositions from old ones in mathematics too.

Definition: If P and Q are proposition then P^Q is a new proposition which referred to as the conjunction of P and Q. The proposition P^Q is true if and only if both P and Q are true propositions.

P Q P Q

T T T

T F F

F T F

F F F


Conjunction (logical product or multiplication)

A conjunction is only true when both p and q are true. Otherwise, a conjunction of two statements will be false:

Truth table:

p q p q

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


Conditional statement To understand the logic behind the truth table for the conditional statement,

consider the following statement.

“If you get an A in the class, I will give you five bucks.” Let p = statement “ You get an A in the class” Let q = statement “ I will give you five bucks.” Now, if p is true (you got an A) and I give you the five bucks, the truth value of p q is true. The contract was satisfied and both parties fulfilled the

agreement. Now, suppose p is true (you got the A) and q is false (you did not get the five

bucks). You fulfilled your part of the bargain, but weren’t rewarded with the five bucks.

So p q is false since the contract was broken by the other party. Now, suppose p is false. You did not get an A but received five bucks anyway. (q is

true) No contract was broken. There was no obligation to receive 5 bucks, so truth value of p q cannot be false, so it must be true.

Finally, if both p and q are false, the contract was not broken. You did not receive the A and you did not receive the 5 bucks. So p q is true in this case.


Truth table for conditional

p q p qT T TT F FF T TF F T


Variations of the conditional

Converse: The converse of p q is q p

Contrapositive: The contrapositive of p q is

┐q ┐p


Examples

Let p = you receive 90% Let q = you receive an A in the course p q ?

If you receive 90%, then you will receive an A in the course. Converse: q p

If you receive an A in the course, then you receive 90% Is the statement true? No. What about the student who receives a

score greater than 90? That student receives an A but did not achieve a score of exactly 90%.


Example 2 State the contrapositive in an English sentence: Let p = you receive 90% Let q = you receive an A in the course p q ? If you receive 90%, then you will receive an A in the course

┐q ┐p If you don’t receive an A in the course, then you didn’t receive 90%. The contrapositive is true not only for these particular statements but

for all statements , p and q.


Logical equivalent statements

Show that is logically equivalent to

We will construct the truth tables for both sides and determine that the truth values for each statement are identical.

The next slide shows that both statements are logically equivalent. The red columns are identical indicating the final truth values of each statement.

p q p q


Documents

1 Introduction to Abstract Mathematics The Logic and Language of Proofs Instructor: Hayk Melikya [email protected] Proposition: Informal introduction to