Proof by Contradiction in Mathematics

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Questions and Answers

Which of the following statements correctly summarizes the proof by contradiction used in the example about the square of an even integer?

The proof starts by assuming that the square of an even integer is odd and then arrives at a contradiction, proving that the square of an even integer must be even. (correct)

The proof assumes that the square of an even integer is even and arrives at a contradiction, proving that the square of an even integer must be odd.

The proof shows that there is no contradiction in assuming the square of an even integer can be both odd and even.

The proof demonstrates that the square of an even integer can be both odd and even, leading to a contradiction.

In the proof that √2 is irrational, what is the contradiction that is reached?

The proof shows that √2 can be expressed as a fraction of two integers, contradicting the initial assumption that it is irrational.

The proof assumes √2 is rational and then arrives at a contradiction, showing that √2 cannot be expressed as a fraction of two integers. (correct)

The proof demonstrates that √2 can be both rational and irrational, leading to a contradiction.

The proof shows that √2 cannot be expressed as a fraction of two integers, contradicting the initial assumption that it is rational.

In the proof that log₂5 is irrational, what is the main reason why the left-hand side (LHS) is odd and the right-hand side (RHS) is even?

The LHS is the logarithm of an odd number to the base 2, which is always odd, while the RHS is an integer, which is always even.

The LHS is multiplied by an odd number (q), while the RHS is multiplied by an even number (2).

The LHS is the result of a logarithmic operation, which always yields an odd number, while the RHS is the result of multiplying an integer by 2, which always yields an even number.

The LHS involves a power of 5, which is always odd, while the RHS involves a power of 2, which is always even. (correct)

What is the key assumption made at the start of the proof about the rationality of log₂5?

log₂5 is a rational number and can be expressed as a fraction of two integers. (B) Signup and view all the answers

In the example demonstrating that if 𝑛2 − 1 is even, then 𝑛 is odd, what is the contradiction that is reached?

It is assumed that 𝑛 is even and then a contradiction is reached, proving that 𝑛 must be odd. (A) Signup and view all the answers

Which of the following statements accurately describes the underlying principle used in all of the examples provided?

All examples demonstrate that if an assumption results in a contradiction, then the opposite of that assumption must be true. (C) Signup and view all the answers

What is the key step taken in the proof that 5 + 7 < 5, which leads to a contradiction?

The proof first tries to prove that 5 + 7 ≥ 5 and then proceeds to show a contradiction, demonstrating that the initial assumption is false. (B) Signup and view all the answers

In the example involving the sum of 𝑎 and 𝑏, what is the contradiction reached?

The proof assumes that 𝑎 + 𝑏 ≤ 5 and then leads to a contradiction, demonstrating that either 𝑎 or 𝑏 must be less than or equal to 2. (A) Signup and view all the answers

What is the remainder when $3^k$ is divided by 2 for any integer k?

1 (C) Signup and view all the answers

Which expression represents the divisibility of $32^n - 1$ by 4?

$32^n - 1 = 4k$ (B) Signup and view all the answers

Which condition is necessary for the expression $a^3 - a + 1$ to be considered odd?

a is a positive integer (D) Signup and view all the answers

What is the final simplified form of $LHS - RHS = a^2 - b^2 + 2b - 2a$?

$0$ (B) Signup and view all the answers

What does the expression $2k + 1$ indicate in terms of number properties?

It is an odd integer (A) Signup and view all the answers

What does the implication $P \Rightarrow Q$ signify?

If $P$ is true, then $Q$ must be true. (C) Signup and view all the answers

Which of the following statements describes $P \Leftarrow Q$?

$Q$ implies $P$. (D) Signup and view all the answers

In the context of integers, which statement is true regarding $P$ and $Q$ where $P$: $n$ is a positive integer and $Q$: $n$ is an even number greater than 0?

$Q \Rightarrow P$ is false. (D) Signup and view all the answers

Which quantifier correctly fills in the blank: '____ real numbers $x, x^2 \geq 0$'?

For all (B) Signup and view all the answers

What does the expression '$\exists x$ for which $x^2$ is odd' imply?

Some integers have an odd square. (B) Signup and view all the answers

Which of the following implications is false regarding the relationship between $P$ and $Q$?

$P$ is false, therefore $Q$ is false. (B) Signup and view all the answers

Considering the definitions of implication, which of the following is a true statement?

An integer can be positive without being even. (B) Signup and view all the answers

Choose the correct interpretation of '$\forall x, x^2\geq 0$'.

All values of $x$ have a non-negative square. (C) Signup and view all the answers

What is the first case to prove when showing that 𝑃 ⇔ 𝑄?

𝑃 ⇒ 𝑄 (D) Signup and view all the answers

In the example provided, if 𝑥 is even, what expression represents 𝑥?

𝑥 = 2𝑘 (C) Signup and view all the answers

What is required to disprove a universal statement?

Provide a counterexample (A) Signup and view all the answers

What method can be used to show a statement is false through contradiction?

Proof by Contradiction (D) Signup and view all the answers

What does proof by contrapositive demonstrate in the example provided?

If 𝑥² is odd, then 𝑥 is odd. (C) Signup and view all the answers

To prove a statement of the form 𝑃 ⇔ 𝑄, how should the two cases be organized?

Under two distinct headings (D) Signup and view all the answers

If a statement applies to at least one number, what symbol represents this quantifier?

∃ (there exists) (A) Signup and view all the answers

In the example provided, what statement is proven by showing if 𝑥 is even, then 𝑥² is even?

If 𝑥² is odd, then 𝑥 must be odd. (C) Signup and view all the answers

What conclusion can be made about the statement 'If 𝑎 − 𝑏 > 0, then 𝑎2 − 𝑏2 > 0' when 𝑎 = −2 and 𝑏 = −3?

The statement is false since it results in a negative value. (C) Signup and view all the answers

Which of the following statements is false regarding prime numbers?

All prime numbers must be odd. (C) Signup and view all the answers

What does the statement '∃ a real number 𝑥, −𝑥2 + 2𝑥 − 2 ≥ 0' imply?

The inequality has no real solutions. (C) Signup and view all the answers

What criterion makes a number divisible by 6?

It is divisible by both 2 and 3. (C) Signup and view all the answers

What can be inferred about the sum of two integers in relation to their parity?

The sum is even if both integers are either odd or even. (C) Signup and view all the answers

Which of the following statements about the divisibility by 4 is accurate?

A number is divisible by 4 if the last two digits form a multiple of 4. (D) Signup and view all the answers

How can a three-digit number be determined as divisible by 9?

If the sum of its digits is divisible by 9. (A) Signup and view all the answers

What can be concluded about the product of two irrational numbers?

The product can be either rational or irrational. (D) Signup and view all the answers

What inequality holds true for $a > b > 0$?

$a^2 - ab + ab - b^2 ext{ is greater than or equal to } (a + b)(a - b)$ (D) Signup and view all the answers

If $x$ is the smallest side of a triangle with sides $x$, $10$, and $12$, what must be true?

$x > 2$ (A) Signup and view all the answers

What can be concluded from $a - b + c - b ext{ for } a > b > c > 0$?

$a - b + c - b ext{ is greater than or equal to } a - c$ (B) Signup and view all the answers

What is the correct conclusion regarding $x^4 + y^4$ and $x^3y + xy^3$ for $x, y > 0$?

$x^4 + y^4 ext{ is greater than or equal to } x^3y + xy^3$ (D) Signup and view all the answers

What does the property of a square of a real number state?

The square is always non-negative (B) Signup and view all the answers

Which statement is true regarding the expression $a^2 + b^2 + c^2 - ab + bc + ca$ for any real numbers $a, b, c$?

It is always non-negative. (A) Signup and view all the answers

If $x$ is the longest side in a triangle with sides $x$, $10$, and $12$, what condition must $x$ satisfy?

$x < 22$ (A) Signup and view all the answers

In the context of inequalities, what can be inferred by applying the triangle inequality?

The inequality sign always stays the same. (B) Signup and view all the answers

What happens when two negative numbers are multiplied?

The result is positive. (B) Signup and view all the answers

Which expression represents the fact that $x^2$ is always greater than or equal to $0$?

$orall x ext{ in } ℝ, x^2 ext{ is non-negative}$ (A) Signup and view all the answers

Study Notes

HSC Mathematics Extension 2

Textbook author is Steve Howard
Textbook is for the 2017 NSW Syllabus
Textbook was last updated in December 2021
Check howardmathematics.com for the latest version
Copyright remains with the author
Licence granted for non-commercial student and teacher use
Commercial use prohibited
Permission required for non-registered schools and organisations
Publication is independent and not affiliated with NESA or the NSW Department of Education

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Description

This quiz explores various proofs by contradiction, particularly those involving the irrationality of numbers and properties of integers. Participants will analyze key assumptions and contradictions reached in different mathematical statements. Test your understanding of these fundamental concepts!