Proof by Contradiction in Mathematics

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

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

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?

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

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

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

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

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

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

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

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

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

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

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

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

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?

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

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

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

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

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

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

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

What is required to disprove a universal statement?

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

What does proof by contrapositive demonstrate in the example provided?

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

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

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

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

Which of the following statements is false regarding prime numbers?

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

What criterion makes a number divisible by 6?

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

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

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

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

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

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

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

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

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

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

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

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

What happens when two negative numbers are multiplied?

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