Exam 14: Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

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(a) The graph $K _ { 2,5 }$ has two vertices of degree 5 and five vertices of degree 2 . Hence it has an Euler path and no Euler circuit.
(b) There is no Hamilton path in this graph since any path containing all five vertices of degree 2 must visit some of the vertices of degree 5 more than once.

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(a) There are $4 ^ { 3 } = 64$ functions from a set with three elements to a set with four elements.
(b) There are $4 \cdot 3 \cdot 2 = 24$ one-to-one functions from a set with three elements to a set with four elements.
(c) There are no onto functions from a set with three elements to a set with four elements.

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The basis step follows since $1 ! = 1 = 2 ^ { 0 }$ . For the inductive hypothesis assume that $n ! \geq 2 ^ { n - 1 }$ . Then $( n + 1 ) ! = ( n + 1 ) \cdot n ! \geq ( n + 1 ) \cdot 2 ^ { n - 1 } \geq 2 \cdot 2 ^ { n - 1 } = 2 ^ { n }$ .