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- Give the names of the following (lower case) Greek letters: ฯ , ฯ. Write the lower case Greek letters theta and rho. [4 marks]
For each of the following sets S, give a function f (n) such that S = {f (n) | n โ N}.
(Recall that N = { 0 , 1 , 2 , 3 , 4 ,.. .}.) a) S = {โ 18 , โ 8 , 2 , 12 , 22 , 32 ,.. .}. b) S = {m โ N | m > 0 and m is a multiple of 4}. c) S is the set of all even integers, that is S = {.. .โ 6 , โ 4 , โ 2 , 0 , 2 , 4 , 6 ,.. .}. [12 marks]
- Negate each of the following statements: a) a = 1 or a โฅ 2. b) x > y > z. c) If f (x) < f (y) then x < y. d) โN โ N, โx โ R, f (x) < N and g(x) < N. [12 marks]
Definition: Let f (x) be a (real-valued) function. Then f (x) is injective if for all x, y in the domain of f ,
f (x) = f (y) =โ x = y.
a) Write down what it means for f not to be injective. b) Prove that the function f (x) = 10 โ 3 x, x โ R is injective. c) Prove that the function f (x) = x^2 + x, x โฅ โ1 is not injective. d) Prove that the function f (x) = x^2 + x, x โฅ 0 is injective.
[13 marks]
Paper Code MATH 104 Page 2 of 4 CONTINUED
- Write down carefully the meaning of the statement that m|n (โm divides nโ), where m and n are integers.
Definition: Let R be a relation on a set X. Then R is an equivalence relation if for all x, y, z โ X the following three conditions hold:
i) x R x. ii) If x R y then y R x. iii) If x R y and y R z then x R z.
Determine whether or not the following relations R on the given sets X are equivalence relations. You should justify your answers carefully, working directly from the definitions. a) X = Z, x R y if x 6 = y + 1. b) X = Z, x R y if 10|(x โ y). c) X = {n โ N | n โฅ 2 }, x R y if there is a prime number which divides both x and y. (Recall that a prime number is an element of the same set X which has no factor other than itself and 1.) [14 marks]
a) Prove the following proposition. Let m and n be integers. If m and n are odd then mn is odd. State the converse of this proposition and also state, with a proof, whether the converse is true. b) Prove the following proposition: There do not exist integers m and n such that 6m + 9n = 22. c) Let P be the following proposition: Let a, b โ R. If a 6 = b then (a + b)^2 > 4 ab. State and prove the contrapositive of P. [15 marks]
Paper Code MATH 104 Page 3 of 4 CONTINUED
