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Answer the question 15, 17, 20, 21 with full steps

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15. Given A € P(X) define the characteristic function xx: X – {0,1} by

XA(x) =

0 if IA,

1 ifre A.

Suppose that A and B are subsets of X.

(i) Prove that the function x + XA(x)XB(x) (multiplication of integers) is the characteristic function

of the intersection An B.

(ii) Find the subset C whose characteristic function is given by

xo(I) = XA(I) + XB(1) – XA(3)x8(I).

16. Determine which of the following functions f: R-R are injective, which are surjective and which

are bijective. Write down an inverse function of each of the bijections.

(i) f(x) = x – 1;

(ii) f(x) = x};

(iii) f(x) = x3 – X;

(iv) f(x) = x3 – 3×2 + 3x – 1;

(v) f(x) = ex;

>0,

(vi) fole) = { *** ifike.

(2

17. Functions f. RR and g: RR are denned as follows.

f(x) =

I +2 if I < -1,
if -11,
I-2 if : > 1.

1-2 if 1.

g(x) =

Find the functions fog and gof. Is g the inverse of the function f? Is f injective or surjective? How

about g? Sketch and compare the graphs of these functions.

18. Suppose that f: X – Y and g: Y Z are surjections. Prove that the composite g of: X Z is a

surjection.

19. Let f: X – Y be a function. Prove that there exists a function g: Y-X such that fog = Iy if and only if

f is a surjection. [g is called a right inverse off.]

20. Let f: X Y be a function and A1, A2 € P(X).

(i) Prove that A, S A2 + 7 (41) S 7 (A2). Prove that the converse is not universally true. Give a simple

condition on f which is equivalent to the converse.

(ii) Prove that 1 (An A2) ST (A) 7 (A2). Prove that equality is not universally true.

(iii) Prove that 7 (A, U A2) = 7 (A) U T (A2).

21. Let f: X – Y be a function. Prove that

(i) fis injective is injective – T is surjective,

了

(ii) fis surjective – is surjective + F is injective.

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