Question

Discrete Math - Please be detailed. Thanks!

. Below is one of the classic fallacies. Note that each step is justified. This is the amount of details we would like to see in your proofs. Identify the fallacious step and explain. 5 points STEP 1: Let ab. STEP 2: Multiply both sides by a, we get a2 ab STEP 3: Add a2 to both sides, we get a2 + a2-ab + a2b STEP 4: Collecting like terms, we get 2a2 ab+a2b STEP 5: Subtract 2ab from both sides, we get 2a2-2ab aba2b-2ab STEP 6: Collecting like terms, we get 22-2ab ab ab. STEP 7: Factoring out constants from both sides, we get 2(a2b-aba2b ab) STEP 8: Cancelling out (ab- ab) from both sides, we get 2 5. Negate the following statements. 5 points cach (a) The function f(r) has a constant derivative and it is linear. (b) If n is even, then 끌 is an integer (c) The number is even or the number y is even. 6. Prove the following statements using direct proof. 10 points cach (Remember this course is about learning how to read/write formal proofs! Use formal definitions in your proofs all the time including the formal definitions of odd and even numbers! "Odd times odd is odd", "odd+odd-even" type of arguments are not part of our formal proofs!) (a) Let a be an odd integer. Use formal definition of odd integers to prove that a2 + 2a + 1 is even Note that following is NOT a proof of this statement Let a 3, then 32+2(3) +1-9+6+1 16 is even." (b) Suppose a,b,ceZ. Prove that if a2lb and blc, then a e. (c) Prove by using the cases that if r E R, then r+3 -r> 2

Answer 1