b) Decide whether the following are true or false giving a proof or cour terexample as appropriate. (i) For all a, n, m E Z. If a | n and a | m then a | (n + m). (ii) For all a, b, n, m E Z. If a |n and b | m then (a + b) | (n +m).

Q: Prove: | a | - | b | <= | a + b |

A:

Q: Show that if a < c< b, then fave (a, b) = () fave l(a, c) + C - a | favg [c, b]

A: Solution...

Q: We assume that x0. Therefore there exists a natural number n such that n> Since z is fixed we can…

A: Suppose that x and y are 2 distinct real numbers. We have to prove that, there exists a rational…

Q: Solve 2|7-x| 8 c. -8 6

A: We have given the inequality; 2|7-x| &lt; 2

Q: For all n, d EZ where d + 0, d\n n = dk for some kEZ. Disprove: For all integers a, 6, and c, if…

A: I have disproved the result

Q: Deduce from Q2 that if an integer N can be written as N = x² + ny² , and if a prime number q | N can…

A: We will use standard division algorithms to deduce the given statement.

Q: (a) Show that if a > 0, then 1/a> 0 and 1/(1/a) = a. (b) Show that if a <b, then a < (a+b) < b.

A:

Q: Use a proof by contraposition to show that if x + y ≥2, where x and y are real numbers, then x ≥1 or…

A:

Q: . Consider the statement: Let A, B ∈ Z. If 6|A and 10|B, then 4|(A2 + B2). (a) Prove the statement.…

A: 5. Consider the statement: Let A, B ∈ Z. If 6|A and 10|B, then 4|(A^2 + B^2). (a) Prove the…

Q: Which choice gives an example that supports the conjecture, and a counterexample that shows the…

A: The objective is to select the correct option from the given options. Given that, for any real…

Q: The sum of the terms 1 for k= 3, 4, ... 10. Provide your answer below: || ])

A: The series given is  3+4+5+6+7+8+9+10 Now asked to find summation notation of the given series…

Q: Given Q(x, y) means “x and y are real numbers such that x + 2y = 5", evaluate 3xvy Q(x, y) O True…

A:

Q: Let A = {1,2} and B = {3,4}. Prove or disprove the statement: “ If C = A × B , then |C| = 4 "

A: We will use the concept of multiplication of 2 sets and then we will find the cardinality.

Q: P.3 Evaluate d dx**+2x]

A: Given that, ddxx2+2x we have to differentiate given function with respect to x

Q: 3. Prove or disprove: If a | (b + c), then either a |b or a | c. h of the follo

A: Since you have asked for question number 3 , we have solved it for you.

Q: Suppose a, b, c, d e Z. If a | b and c| d, then ac | bd. а

A:

Q: 5. Evaluate the following expressions: A. (10+j6) + 10 Z-30 - 10/30 5 - j10 j4 B.62 - 120 + C. 10+j5…

A: Since you have posted a question with multiple sub-parts, we will provide the solution only to the…

Q: Suppose that P(A) = 0.3 and P(B) = 0.2 .(a) What does P(A u B) equal if 1. A and B are mutually…

A:

Q: Draw the graphs of the inequalit a.) y> 2x – 6 -

A: Given query is to draw the graph of the inequality.

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.