CSL 101 Discrete Mathematics Lecture 3-4

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Questions and Answers

What is the base case for proving P(n) in the tiling problem?

Tiling a 2x1 puzzle

Tiling a 1x1 puzzle

No tiles needed for a 0x0 puzzle (correct)

Tiling a 2x2 puzzle

What does Theorem B state about tiling with Bill?

Bill is not required in a 2n x 2n puzzle.

Bill can be placed anywhere in a 2n x 2n puzzle. (correct)

Bill cannot be placed in any 2n x 2n puzzle.

Bill can only be in the middle of the puzzle.

In the induction step, what is assumed to be true for the proof?

Bill can be placed anywhere in 2n x 2n puzzles. (correct)

Bill can only be tiled in corners.

Tiling is impossible for odd dimensions.

Tiling can only be done with 4 tiles.

What is a new idea introduced in the proof for the tiling problem?

Grouping squares together. Signup and view all the answers

Why is it important to prove that the number of squares is divisible by 3?

To validate the induction step. Signup and view all the answers

What does Theorem 2 state about an infinite subset of a countably infinite set?

The infinite subset is countable. Signup and view all the answers

Which proof technique involves proving that the negation of the conclusion leads to a contradiction?

Proof by contradiction Signup and view all the answers

What characterizes an even integer?

It can be expressed as 2k. Signup and view all the answers

Which of the following is an example of direct proof for a statement?

Assuming A is true and showing B must also be true. Signup and view all the answers

In the proof regarding the closed interval [0,1], what is concluded about listing the real numbers?

It contradicts the assumption of a bijection from N to [0,1]. Signup and view all the answers

According to the proofs presented, what can be said about the product of two odd integers?

It is an odd number. Signup and view all the answers

How is the function g defined to show it is a bijection from N to an infinite subset B of a countably infinite set A?

By inductively choosing the smallest unselected element from A that belongs to B. Signup and view all the answers

What is the result when adding two even integers, according to the provided proofs?

The result must be even. Signup and view all the answers

What principle is used to prove statements in mathematical induction?

Prove a base case and then use an assumption for the next case Signup and view all the answers

Which of the following sequences does mathematical induction apply to?

Any non-negative integer sequence Signup and view all the answers

In the inductive proof structure, which statement is correct about P(0)?

It must be proven unconditionally Signup and view all the answers

Which statement correctly describes the induction step in mathematical induction?

Assume P(n) is true and prove P(n + 1) Signup and view all the answers

What happens in the case of an integer n that is not prime according to the logic presented?

It implies n is divisible by a smaller number. Signup and view all the answers

Which of the following best describes the outcome of the induction process?

It establishes a general truth for all natural numbers. Signup and view all the answers

What is the overall significance of the induction hypothesis in the proof process?

It allows proving subsequent cases based on previous results. Signup and view all the answers

How does mathematical induction illustrate its effectiveness?

Through a domino effect of proving successive cases. Signup and view all the answers

What is the first step in the induction proof process?

Establishing the base case Signup and view all the answers

In the induction step, what must be demonstrated?

P(n) implies P(n+1) Signup and view all the answers

Which of the following is NOT a valid component of a proof by induction?

Conclusion for n = 2 only Signup and view all the answers

Which case could be used to prove divisibility by 6 using induction?

Assuming it is divisible by 2 and proving both Signup and view all the answers

What is the goal when proving an implication of the form 'If P, then Q'?

Prove the contrapositive: 'not Q implies not P.' Signup and view all the answers

Which proof method is used to prove that an integer is even if and only if its square is even?

Constructing a chain of if and only if statements Signup and view all the answers

What does UG stand for in the context of induction?

Universal Generalization Signup and view all the answers

In proving the contrapositive of 'If m is even, then m² is even,' which statement represents the contrapositive?

If m is odd, then m² is odd. Signup and view all the answers

For which integer n is the base case true in the context of proving an inequality?

n = 2 Signup and view all the answers

In the context of the induction process, which is most critical to establish?

Demonstrate that a property holds for all integers Signup and view all the answers

Which of the following is a key characteristic of a proof by contradiction?

It assumes the opposite of the statement is true. Signup and view all the answers

What is the significance of the assumption P(i) in the induction step?

It serves as the foundation to show P(i + 1) Signup and view all the answers

What is the conclusion drawn from the proof of the statement that 'If √r is rational, then r is rational'?

If √r is rational, then r must be rational. Signup and view all the answers

In the context of proving that 2 is irrational, what is established by choosing integers m and n without common prime factors?

It sets up a condition for finding a common factor. Signup and view all the answers

Which of the following correctly represents a direct proof approach?

Showing that the implication holds through established relationships. Signup and view all the answers

To prove that 'If P, then Q' is equivalent to 'If not Q, then not P', what type of proof method is utilized?

Contrapositive proof Signup and view all the answers

Study Notes

Countability

A countably infinite set A ensures any infinite subset B of A is also countable.
A bijection f: N → A allows for the construction of a bijection g: N → B through inductive reasoning.

Uncountable Sets

The closed interval [0,1] is proven uncountable using a hypothetical bijection r: N → [0,1].
A unique number exists that cannot be listed in the assumed bijection, demonstrating a contradiction.

Basic Proof Techniques

Various logical proof strategies include direct proof, contrapositive, proof by contradiction, and proof by cases.

Direct Proofs

An even integer n can be expressed as n = 2k; an odd integer n can be expressed as n = 2k + 1.
The sum of two even numbers is shown to be even through algebraic manipulation.
The product of two odd numbers results in an odd number, demonstrated stepwise.

Proving Implications

To prove "If P, then Q," direct proof involves assuming P and logically showing Q follows.
Proving the contrapositive involves demonstrating that "not Q implies not P."

Logical Equivalence

An integer is even if and only if its square is even, needing proof in both directions for validation.
Utilizing contrapositive reasoning allows alternate paths to establish logical equivalence.

Proof by Contradiction

The irrationality of √2 is established by supposing it is rational and resulting in a contradiction.
The prime factor consideration method concludes that every integer greater than 1 must possess a prime number.

Mathematical Induction

Induction involves proving a base case (usually P(0)), then assuming P(n) is true to show P(n+1) must also be true.
This creates a domino effect leading to the conclusion for all integers.

Induction Step Example

When tasked to prove a general statement via induction, the process includes establishing a base case and logical progression to the next integer.
Inductive reasoning is employed in various proofs related to divisibility and properties of integers.

Puzzle and Tiling Theorem

A tiling puzzle demonstrates the intriguing application of induction to show how any 2n x 2n puzzle can be covered with L-shaped tiles while leaving one square empty for "Bill".
The theorem reaffirms this capability when generalized, allowing placement of Bill in any square rather than just the center.

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Description

This quiz covers key concepts from Lecture 3-4 of the CSL 101 Discrete Mathematics course, specifically focusing on countability and Theorem 2 regarding countably infinite sets. Students will explore the implications of bijections and subsets in the realm of set theory.