Math Worksheet

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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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