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# MTH 4150: Final Practice 3

Problem 1

Let $$A=\left\{1,2,3,4,5\right\}$$ and $$B=\left\{5,6,9,11\right\}$$. Provide example of two functions $$f:A\to B$$ and $$g:A\to B$$ with domain $$A$$ and codomain $$B$$ such that $$f$$ is onto but $$g$$ is not onto.

Problem 2

Let $$A=\{1,2,3,4,5,6,7\}$$. Provide an example of a permutation $$\sigma$$ on $$A$$ that satisfies all of the following three conditions:

• $$1^{\circ}$$ The permutation $$\sigma$$ has one cycle of length 4 and one cycle of length 3;

• $$2^{\circ}$$ The number $$1$$ is sent to a number greater than $$4$$, i.e. $$\sigma(1) > 4$$; and

• $$3^{\circ}$$ The number $$7$$ is sent to a number smaller than or equal to $$5$$, i.e. $$\sigma(7)\leq 5$$.

Problem 3

Let $$F(X)$$ be the generating function of the sequence $$(a_n)_{n=0}^{\infty}$$ defined as $$a_n=n$$ for $$n\in\{0,1,2\}$$, and $$a_n=5$$, for $$n\geq 3$$. Calculate $$F\left(\frac 13\right)$$.

Problem 4

A box contains $$30$$ red, $$40$$ green, and $$50$$ blue marbles.

• (a) What is the smallest number of marbles that a blindfolded person has to take from the box to be sure that at least three marbles are of the same color?

• (b) What is the smallest number of marbles that a blindfolded person has to take from the box to be sure that at least three marbles of different colors are taken?

Problem 5

Determine the number of ways in which the faces of the cube can be painted in $$5$$ colors in such a way that not all faces are of the same color. The paintings that can be obtained from each other using rotations are considered identical.

Problem 6

Initially the numbers $$10$$, $$15$$, and $$20$$ are written on three different whiteboards. Two players $$A$$ and $$B$$ play the following game. The player $$A$$ starts and players alternate the moves. In each of the moves, a player has to choose one of the whiteboards, erase the number written on the board and replace it by a bigger two-digit number. The player who makes the last move is the winner. Prove that $$A$$ has the winning strategy. What should player $$A$$ do in its first move?

Problem 7

Let us denote by $$A$$ the set of all positive integers smaller than or equal to $$10$$. Find the number of functions $f:A\to\left\{1,2,3,4,5\right\}$ for which the set $$B=\left\{x\in A: f(x)=5\right\}$$ has an odd number of elements.

Problem 8

Let $$U=\left\{1,2,3,\dots, 100\right\}$$ and let $F=\left\{ (A,B,C): A, B, C\subseteq U\mbox{ such that } A\subseteq B\subseteq C\subseteq U\right\}.$ (In words, $$F$$ is the sequence of all triplets whose components are subsets of $$U$$ such that the first component is a subset of the second, and the second is the subset of the third.) Calculate $$|F|$$.