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- Give the names of the following (lower case) Greek letters: ฯ , วซ. Write the lower case Greek letters beta and phi. [8 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 10 , โ 3 , 4 , 11 , 18 , 25 ,.. .}. b) S = { 4 , 8 , 16 , 32 , 64.. .}. c) S = Z = {... โ 3 , โ 2 , โ 1 , 0 , 1 , 2 , 3 ,.. .}. [12 marks]
- Negate each of the following statements: a) 2 x + 5y = 6. b) โ 1 โค a < 1. c) If x = y + 2ฯ then f (x) = f (y). d) โn โ N, โx โ R, f (x) > n. [12 marks]
Definition: Let f (x) be a (real-valued) function. Then f (x) is injective if for all x, y โ R, f (x) = f (y) =โ x = y.
a) Write down what it means for f not to be injective. b) Determine whether or not the function f (x) = 20โ 5 x is injective. You should justify your answer carefully, working directly from the definition. c) Solve the equation x^2 + x = 0 where x โ R. Hence show that the function f (x) = x^2 + x is not injective.
[10 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 = R, x R y if 2x โฅ y. b) X = Z, x R y if 6|(x โ y). c) X = Z, x R y if x + y is even. [14 marks]
- Write proofs of the following statements. You should work from the defi- nitions:
Definition: Let n โ Z. Then (i) n is even if there exists an integer k such that n = 2k. (ii) n is odd if there exists an integer k such that n = 2k + 1.
a) Let m, n โ Z. If m is even and n is odd then m + n is odd. b) If n โ Z is odd then n^2 is odd, and if n โ Z is even then n^2 is even. c) It is impossible for the square of an integer n to be of the form 4m + 2 for an integer m. (Hint: consider separately the cases where n is even and n is odd.) [15 marks]
Paper Code MATH 104 Page 3 of 4 CONTINUED
