Statement: No integer is both even and odd.

For the following statements identify the mistakes in the beginning of the proof. Prove: For all integers a and b, if a and b are divisible by 3 then a + b is divisible by 3. Beginning of proof: Suppose that for all integers and b, if a and b are divisible by 3 then a + b is divisible by 3. By definition of divisibility…   Prove by contradiction: The product of any irrational number and any rational number is irrational. Beginning of proof: Suppose not. That is, suppose the product of every irrational number and every rational number is rational...

Prove this statement by contraposition: For all real numbers ?, if 2?3 − 5?2 + 11? + 7 is irrational then r is irrational.

What is the pattern to the results of the following multiplications ? 1 x 1 11 x 11 111 x 111 1111 x 1111 After describing the pattern, use inductive reasoning to find the value 111,111,111 x 111,111,111 without  actually doing the direct multiplication.

Give a proof by contraposition of the following statement. If x is an integer and square of x is even, then x is even.

Use the given information to provide a direct proof of the statement below. Prove that the product of two odd numbers is an odd number. Definition: An odd number is an integer n that can be written in the form n Fact 1: If a and b are integers, then so are a + b and a · b. = 2k +1 for some integer k.

Give a counterexample to the statement. The product of 2 primes is always odd.

What is the inverse of the proposition: “If 2 is an even number, then 2 is rational.” ?

Discrete Math; prove by contraposition: Negate state and disprove the original statement

Please answer the following questions.