10. [6] Two integers are consecutive if, and only if, one is one more than the...
4. Two integers are consecutive if, and only if, one is one more than the other. Prove the following statement: The difference of the squares of any two consecutive integers is odd. [8 points]
the larger of two consecutive integers is 12 more than twice the smaller. what is the inequality and what are the integers.
6. (20 points) Problem 2, page 91. Prove that the sum of two even integers is even. Use the three proofing techniques (a) a direct proof (b) a proof by contradiction (c) a proof by contraposition
The sum of the two consecutive odd integers is sixteen more than the four times the smaller integer. Find the integers.
Find two consecutive odd integers such that 32 more than the lesser is three times the greater The lesser consecutive odd integer is and the greater consecutive odd integer is
three consecutive even integers are such that the sum of the smallest and 3 times the second is 38 more than twice the third. what are the integers?
Question 4 10 marks You should be able to answer this question after studying Unit 9. (a) Use proof by contraposition to prove that the following statement is true for all integers n: If n2 + 2n + 7 is odd, then n is even. [6] (b) Hence prove that the following statement is true for all integers n: na + 2n + 7 is odd if and only if n is even. [4]
1. Consider the following claim. Claim: For two integers a and b, if a + b is odd then a is odd or b is odd. (a) If we consider the claim as the implication P =⇒ Q, which statement is P and which is Q? (b) Write the negations ¬P and ¬Q. (c) (1 point) Write the contrapositive of the claim. (d) Prove the contrapositive of the claim. 2. Use contraposition (proof by contrapositive )to prove the following claim....
Word Problem: Find the lesser of two consecutive integers whose sum is greater than 16.
Assignment 6 1. Prove by contradiction that: there are no integers a and b for which 18a+6b = 1. 2. Prove by contradiction that: if a,b ∈ Z, then a2 −4b ≠ 2 3. Prove by contrapositive that: If x and y are two integers whose product is even, then at least one of the two must be even. Make sure that you clearly state the contrapositive of the above statement at the beginning of your proof. 4. Prove that...