Discusses basic properties of real numbers.
Introduces the concept of assumptions in proofs, relating to real numbers.
Let x and y be two real numbers; they are not necessarily integers.
The symbol used for real numbers is ⦚ (double lined R).
Conditional statement: ( P \implies Q )
( P ): ( x + y \geq 2 )
( Q ): ( x \geq 1 ) or ( y \geq 1 )
Assume ( P ) is true.
Required to show ( Q ) is also true.
Example: Start with ( x + y \geq 2 )
Rearrange to show that ( x ) or ( y ) satisfies the conditions.
The contrapositive: If ( P ) is not true, ( P' ) must hold.
( P' ): ( x + y < 2 )
Leads to ( Q' ): ( x < 1 ) and ( y < 1 ) must be true.
Truth table analysis for logic involving arguments:
Where ( P ) and ( Q ) state conditions.
Analysis of when these conditions yield true or false.
Start with assumptions to establish a contradiction.
Use the properties of weekdays:
If there are 22 days considered, at least four must fall into some weeks.
Analyze different arrangements to extract contradictions.
Mathematical proofs can take various forms (direct, contradiction, contrapositive).
Understanding of foundational concepts (like real numbers) is crucial for successful proofs.