Problem 5. Which of the following are true and which are false? 1, {(0))U0=(0,()
ID: 3146327 • Letter: P
Question
Problem 5. Which of the following are true and which are false? 1, {(0))U0=(0,())) 2. 0)U 0) 0, 0)) 4. {0,(0))n {(0),((0))) = {(0)) 5. For sets A, B. and C, AU (B-C) = (A U B)-(A UC). Problem 6, Consider the function g : Z Q defined by g(n) = 1/(n2 + 2). Is g injective? Is g surjective? What is the image of g? Sketch a graph of g. Problem 7. Let g : R2 R be defined by g(x,y) = x + y. Is g surjective? Is g injective? Problem 8. A function f : R R is said to be odd if Vz e R,f(-x) =-f(x). Suppose f, g :R R are odd functions. Is fog and odd function? Problem 9, A function f : R R is said to be even ifVz E R,f(-x) = f(x). Prove or disprove the following vf :R R,3g : R R s.t. (fis odd) (f o g is even).
Explanation / Answer
5.
1. {{}} U = {{}}
The given statement is false.
2. {{}} U {} = {, {}}
The given statement is true.
3. {, {}} {{}, {{}}}
(Look for common elements. There is only one.)
= {{}}
The given statement is false.
4. {, {}} {{}, {{}}}
(Look for common elements. There is only one.)
= {{}}
The given statement is true.
5. If B = C, A U (B - C) = A
But (A U B) - (A U C) = (A U B) - (A U B) = .
The statement is false.
8. Since f and g are odd, f(-x) = -f(x) and g(-x) = -g(x)
f(g(-x)) = f(-g(x)) = -f(g(x))
Therefore, f.g is odd.