Conditional statements. Converse and inverse theorem. Types of proofs. Mathematical induction. Sequences. Lecture 2

Conditional statements.

Let’s train the brain!
If student pass exams then he will receive the scholarships.
Compound statement:
…. Is sufficient condition for ….
Formulate by yourselves:
…. implies ….
…. Is necessary condition for ….

Theorem. Inverse and converse theorem.
Example 2.1
Theorem. If student pass exams then he will receive the scholarships.
Converse Theorem. If student receive the scholarships then he will pass exams.
Inverse Theorem. If student pass exams then he will not receive the scholarships.

Let’s train the brain!
Theorem. If it rains then the ground is wet. Formulate
a) converse theorem
b) inverse theorem
c) converse of inverse theorem
d) inverse of converse theorem
2) Theorem. If a=0 or b=0 then ab=0.
Formulate
a) converse theorem
b) inverse theorem
c) converse of inverse theorem
d) inverse of converse theorem
2) Theorem. If two sides of a triangle are equal, then it is isosceles
Formulate a) converse theorem b) inverse theorem c) converse of inverse theorem d) inverse of converse theorem

Types of proofs.
Theorem. If a=0 or b=0 then ab=0.

Mathematical induction

Example 2.4 Prove that any sum equal or greater than 8 cents may be collected using coins of 3 and 5 cents
BS. Can we collect sum = 8 cents?
AS. Assume that we can collect sum of n cents.
IS. So we have a sum of n cents. Can you offer the way how we can get n+1
cents? Which coins we need to remove and which coins we need to add?

2.1 Sequences of real numbers

Examples of sequences
Arithmetic progression
2) Geometric progression
3) Approximation to π