2. Suppose S and T are nonempty sets of real numbers with the following property: (P) There exists y in T such that for...
clear and clean answer,pls 2. Suppose S and T are nonempty sets of real numbers with the following property: (P) There exists y in T such that for all x in S, y>x. Prove the following (of course, use the standard approach for proving universally and existentially quantified statements): (Q) For all x in S, there exists y in T such that y>x. 2. Suppose S and T are nonempty sets of real numbers with the following property: (P) There...
3. Prove the following consequences of the well ordering principle: (a) For all nonempty sets S that are bounded below (there exists an a ∈ Z such that for all s ∈ S, a ≤ s), there is a smallest element (there exists an a ∈ S so that for all s ∈ S, a ≤ s). (b) For all nonempty sets S that are bounded above (there exists an a ∈ Z such that for all s ∈ S,...
1. Suppose that a function f, defined for all real numbers, satisfies the property that, for all x and y, f(x + y) = f(x) + f(y). (*) (a) (2 points) Name a function that satisfies property (*). Name another that doesn't. Justify your answers. (b) (3 points) Prove that any function that satisfies property (*) also satisfies f(3x) = 3f(x). (c) (5 points) Prove that any function that satisfies property (*) also satisfies f(x - y) = f(x) –...
1. Suppose that a function f, defined for all real numbers, satisfies the property that, for all x and y, f(x + y) = f(x) + f(y). (*) (a) (2 points) Name a function that satisfies property (*). Name another that doesn't. Justify your answers. (b) (3 points) Prove that any function that satisfies property (*) also satisfies f(3x) = 3f(x). (c) (5 points) Prove that any function that satisfies property (*) also satisfies f(x - y) = f(x) –...
5. Describe the following sets of real numbers and find the supremum and infimum of these sets: (a) {x}\x2 – 2 <4€R} (b) {x|x+ 2 +13 – x4<4} (©) {x|x<for all neN} 6. For any two elements x and y of an ordered field, prove that _x+ y + x- x + y - x - y (a) max{x,y}=- (b) min{x,y}=-
(a) Find (22,8,P) and nonempty sets (An)>1 C 8 such that P (liminf An) <liminf (P(Ar)) < limsup(P (A)) <P (limsup An). (b) Given A, B,(A.)21,(B.).>1 C , either prove the following statements or find counterexamples to them. i. limsupA, U limsupB = limsupA, UB.. ii. liminf Anu liminfB = liminfAUB iii. limsupA, nlimsupB. = limsupA, B. iv. liminfa, nliminfB = limin A, B, (e) Prove that probability spaces have the property of continuity from above. (We proved continuity from...
1. Suppose that P is the uniform distribution on [0,1). Partition the interval [0,1) into equivalence class such that x ~ y (x is equivalent to y) if x-y є Q, the set of rational numbers 2. Given 1, by the Axiom of Choice, there exists a nonempty set B C [0,1) such that IB contains exactly one member of each equivalence class. Prove each of the following (a) Suppose that q E Qn [0, 1). Show that B (b)...
Suppose that the functions s and t are defined for all real numbers x as follows. (r)-5r t(r) 3x-2 Write the expressions for (+s)(x) and (-)) and evaluate (t s)X-2). +3)-0 ts(x) s)(x) = (rs)(-2)D
Suppose that the functions s and t are defined for all real numbers x as follows. s(x)=x-6 t(x)= 3x2 Write the expressions for (s – t)(x) and (s·t)(x) and evaluate (s+t)(-3). (s – t)(x) = _______ (s·t)(x) = _______ (s + t)(-3) = _______
Use the definitions: x ∈ S ∩ T iff (x ∈ S) ∧ (x ∈ T) x ∈ S' iff ¬(x ∈ S) S ⊆ T iff (∀x)(x ∈ S → X ∈ T) 1. Prove line-by-line: r → ¬(p → q) assuming that ¬r ∨ ¬q and ¬(q ∧ r) ∧ p 2. A, B, C & A ⊆ B are sets. Prove: BnCgAnC