Child Development Associate

The following post has two assignments namely;

1.Child Development Associate

Develop your Philosophy using the developmentally appropriate approach you use with preschool children. here are a few guided questions: What do you believe about early childhood education? What do you believe about young children....what are their needs and how do they learn best? How do you meet these needs and provide for optimal learning? What are the elements of a good preschool curriculum and an engaging learning environment? What are the characteristics of a good teacher in your opinion, and which ones are more important? Reflect on how a teacher can build positive relationships with families. How would you meet the needs of children with special needs, and how to best embrace diversity i your program.

2.Real and Complex Analysis

1. Let (X, A, µ) be a measure space. (a) Prove that µ (E ∪ F) = µ (E) + µ (F) − µ (E ∩ F) for all E, F ∈ A. [4 marks] (b) Let X be a non-empty set and let A, B be sigma algebras of X. • Prove that A ∩ B is a sigma algebra. [4 marks] • Give an example to show that A ∪ B is not necessarily a sigma algebra. [3 marks] (c) Let X = {1, 3, 5, 7, 9} and A = {{1, 3} , {3}} ⊂ P(X) • Write down the smallest sigma algebra that contains A, and verify that what you have written is a sigma algebra [3 marks] Let B be the collection of all open sets of R. i. Is B a sigma algebra? [1 mark] ii. Let C ⊂ R be a closed set. Prove that C is in the Borel sigma algebra on R. [3 marks] iii. Give an example of a set that is neither open nor closed, yet is in the Borel sigma algebra on R. Demonstrate that the set you provide has these properties. [2 marks] 2. Consider (R, A, µ) where • A = {E ⊂ R | either E is countable or Ec is countable} • µ: A → [0, ∞] is defined by µ (E) = (P x∈E |x| E countable ∞ E uncountable (a) Prove that (R, A, µ) is a measure space. [6 marks] (b) What is µ ([0, 1])? [1 mark] (c) Give an example of an infinite set E ⊂ R such that µ(E) < ∞. [2 marks] 3. Consider the measure space (R,L, µ) where L is the Lebesgue sigma algebra and µ is the Lebesgue measure. For each j ∈ N define Ej = [j, ∞). Show that µ   \∞ j=1 Ej   6= lim j→∞ µ (Ej ). Why does this not contradict Theorem 7.9? [4 marks] 1