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Mappings, Injective and Surjective Functions and Cycles

1. Let f : X -> Y and g : Y -> Z be mappings.
(1) Show that if f and g are both injective, then so is g o f : X -> Z
(2) Show that if f and g are both surjective, then so is g o f : X -> Z.

2. Let alpha = 1 2 3 4 5 and Beta = 1 2 3 4 5
3 5 1 2 4 3 2 4 5 1 . (I couldn't draw ( ) on both sides of these groups.

(1) Write Alpha and Beta into cycles.
(2) Find AlphaBeta and BetaAlpha. Is AlphaBeta = BetaAlpha?
(3) FInd Alpha^2 , Beta^2 and(AlphaBeta)^2
(4) Find Alpha^2Beta^2. Is Alpha^2Beta^2 = (Alpha-Beta)^2?
(5) Fnd Alpha^-1 and Beta^-1.

3. Let Alpha, Beta E S_n . Show that if AlphaBeta = BetaAlpha, then (AlphaBeta)^n =
Alpha^nBeta^n .

I used S_n for S sub n. E for epsilon. Alpha for its symbol and Beta for its symbol.

Solution Summary

Mappings, Injective and Surjective Functions and Cycles are investigated.

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