Download Math 104 Exercise Solutions: Greek Letters, Sets, Functions, Relations, Inequalities and more Exams Mathematics in PDF only on Docsity!
- Give the names of the following (lower case) Greek letters: σ, χ. Write the lower case Greek letters psi and beta. [8 marks]
- State whether or not each of the following numbers: i) − 1; ii) 0; iii) 9.5; iv) 6;
is an element of each of the following sets. For each possible combination of one of i) – iv) with one of a) – c), you should either state explicitly that the given number is an element of the given set, or state explicitly that it isn’t. a) Z. b) {x ∈ R | x > −1 and x^2 < 20 }. c) {^12 n^2 − 3 | n ∈ Z}. [12 marks]
- Negate each of the following statements: a) a > 1 or a < −2. b) x ≤ y ≤ z. c) If x > y then f (x) > f (y). d) ∀x ∈ R, ∃y ∈ R, x > y and f (x) ≤ f (y). [12 marks]
Definition: Let f (x) be a (real-valued) function. Then f (x) is injective if for all x, y ∈ R, f (x) = f (y) =⇒ x = y.
a) Write down what it means for f not to be injective. b) Determine whether or not the function f (x) = 8x+12 is injective. You should justify your answer carefully, working directly from the definition. c) Show that the function f (x) = x^3 − x is not injective. Again justify your answer from the definition.
[10 marks]
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- Write down carefully the meaning of the statement that m|n (‘m divides n’), 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 ∈ 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 x + y > 0. b) X = Z, x R y if 4|(x − y). c) X = { 2 , 3 , 5 , 7 , 11 }, x R y if x|y. [14 marks]
a) Prove the following proposition: Let m and n be integers. If m and n are odd then 3m + 5n is even. State the converse of this proposition and also state, with a reason, whether the converse is true. b) Let P be the proposition: Let m, n ∈ Z. If m + n is odd then m is odd or n is odd. State and prove the contrapositive of P. c) Prove the following proposition: Let a, b ∈ R. If a 6 = b then (a + b)^2 > 4 ab. [15 marks]
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