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3a²n−1
1. Show if satisfy the algebraic axioms:
identity and inverse under addition.
2. Show if satisfy all (communicative, associative, identity, and inverse) the algebraic axioms under both multiplication and addition.
3. Which number set can you find the inverse of integers under multiplication?
4. Which number set can you find the inverse of natural numbers under multiplication?
5. Find the prime factors of the following numbers:
(a) 2003
(b) 1560
(c) 5680
(d) 3050
6. Calculate gcd of the following using Euclidean Algorithm:
(a) (572, 279)
(b) (138, 114)
(c) (578, 255)
(d) (688, 212)
1. Is the function 𝑓: ℤ → ℤ 𝑓(𝑥) = 𝑥 2 + 3 injective, surjective or bijective? Prove your assertions
Provethat12 +32 +52 +⋯+(2n+1)2 = (n+1)(2n+1)(2n + 3)∕3 whenever n is a nonnegative integer.
Find a formula for
1+1+⋯+ 1
1 ⋅ 2 2 ⋅ 3 n(n + 1)
by examining the values of this expression for small
values of n.
b) Prove the formula you conjectured in part
Question 6 (6 marks)
6a) Draw the Venn diagrams for each of these combinations of the sets A, B, and C: (i) (A – C)C ⋂ ( B- C)C (3 marks)
(ii) (A – C) U (C – B)(3 marks)
End
5a) Explain whether each of the following relations on the set of real numbers is a function or not. For those (if any) that are indeed functions say whether they are one-to-one and/or onto. (2 marks)
i) y = f(x) = 2x2+1 xэR, y эR
ii) y = g(x) = 1/(x+1) (xэR, y эR , x != -1)
iii) Let h be a function from X = {1, 2, 3, 4} to Y = {a, b, c, d}.
h(1) = d, h(1) = c, h(2) = a, h(3) =b, and h(4) = b.
5b) Does either f or g have an inverse? If so, find this inverse. (1 marks)
5c) Find the composite functions f 。g and g。f . (2marks)
Question 6
(b) Let A = {5, 6, 7, 8}, B = { 4, 6, 7} and the relations
R1 = {(a, b) | a эA, bэB and a > b}
R2 = {(a, b) | a э A, b э B and (a – b)2 <=6}
(i) Find the sets of ordered pairs in R1, R2 and give their cardinalities |R1|, |R2|. (2 marks)
(ii) Draw the directed graphs of R1, R2. (2 marks)
(iii) Give the Boolean-matrix representations of R1, R2. (2 marks)
(c) Explain what are Reflexive, Symmetric relations using examples. Each relation should contain at least three elements. (2 marks)
(a) Let A = {0,1, 2, 3, 4, 5}, B = {1, 2, 3, 4, 5, 6}, and consider the relation
R = {(a, b) э A x B | a2+b2 < 30}.
(i) List all the elements of the relation R and give its cardinality |R|. (3 marks)
(ii) Find the domain and range of the relation R. (2 marks)
(iii) Find the inverse relation R-1 (2 marks)
Let A = {a,b,d,e,g,f,1,2,3}, B = {1,2,3}, and C = {1,2,3,a,d,g}. Find the following:
(i) A ՈB Ս C
(ii) A-B
(iii) (B Ո C) x (A Ո C)
(iv) P(B Ո C)
(v) |CxA|
