5. Negate the following statements. 5 points cach (a) The function f(r) has a co
ID: 3750711 • Letter: 5
Question
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 is an integer. (c) The number z is even or the number y is evern 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,ce Z. Prove that if a2lb and bc, then a lc. (c) Prove by using the cases that if r E R, then 31 -x> 2.Explanation / Answer
Solution:
Note: The first question is done as per Chegg guidelines, please repost others.
(a) Let p : The function f(x) has a constant derivative and q : The function is linear.
The given statement is p ^ q.
The negation of p ^ q according to De-Morgan's law is ~p v ~q.
The answer is The function f(x) does not have a constant derivaitive or is not linear.
(b) Let p : n is even and q : n/2 is an integer.
The given statement is p -> q.
Using material implication, this is equivalent to ~p v q.
The negation according to De-Morgan's law is p ^ ~q.
The answer is n is even and n/2 is not an integer.
(c) Let p : The number x is even and q : The number y is even.
The given statement is p v q.
The negation according to De-Morgan's law is ~p ^ ~q.
The answer is The number x is not even and the number y is not even.
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