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: a || b m. 2 1 4 3. n 6. 8 22 = 10x – 4 and m26 = 5x + 21

A:

Q: a Prove that < 16 – a| for each a, b e R. - V62 + 1 Va? + 1

A:

Q: Ex: Let X₂ [1,2,3,4,5]. I. [0,X, [1], 82.3, 81.23], Y. [3,4,5,6,73 andl T₁: [0, X, [4], [4,633 if A:…

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: Haw:2 ExPress the Sumef the Vectars Aß and E ,if ACl,-), B (2,0)_, C(-1,)_ and D(-2,2).

A: Here is the solution

Q: Let a, b eR with a < b. Assume that a + <b - for all n E N. Show that b- = (a, b). %3D n n=1

A: We will show  two sets are equal.

Q: Find the value of t so that t 4 -4| 2 3 3=4 4 5 0

A:

Q: 11! 1.) 5!x4!x3! 2.) Р(8, 5) 3.) Р(12, 4)

A: Our aim is to solve the following :  1) 11!/(5!*4!*3!) 2) 10!/7! 3) P(8,5) As per our guidelines, we…

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: b.) 3x-2y < 12

A:

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: What is L! if 2 1 +. s s-4] 4r A. -2t +e В. -t+ 2e¬7! 3t +e" D. -2t+e" C. 47 3r E. 4t - 2e"

A:

Q: The solution of 14 - x|>1 is Select one: O A. [-2, 5] O B. ]-00,-5], [-3,00[ OC.J-00,3], [5,00[ O D.…

A:

Q: Let A, B, and C are three sets, prove that Bx(AUC)=(B× A)U(B×C)

A: "Since you have posted multiple questions, we will answer the first question. If you want the…

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: .Prove that la + b + c| < |a] + |b| + ]c]

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: 1. (2ab)3 Ob) 2ab3 Oa) 8a3b3 O d) 6ab O ) 6a b3 Simplify:(x*Xx?) 2. O b) x6 O a) x5 O d) x1.5 O C) X

A:

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: Which of the answers below is the ordering of these running times from fastest to slowest? a)…

A: We have to arrange the following sequences in order from fastest running times to slowest. (a)…

Q: Suppose b1 + *+ bm < n. Prove that b,!bm!< n!.

A:

Q: O Solve The Set of Imear eguatiomos af (2 - 3X - 2y +42= 9. -34-22=5 4X+3y-2Z=7 %3D %3D

A:

Q: If+b+=++, then show that for 1 a" +b" +c"

A:

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: If m|n then lcm(m,n) = n. :Select one True False

A: True

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.