sol-midterm2 - MACM 101 Discrete Mathematics I Outline...

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MACM 101 — Discrete Mathematics I Outline Solutions to Exercises on Functions and Relations 1. Give a definition of a one-to-one function. Let f be a function from A to B . The function f is called one-to-one if and only if for any a,b A f ( a ) = f ( b ) implies a = b . 2. Give a definition of the composition of two functions. Let g be a function from A to B and f be a function from B to C . The composition of functions f and g , denoted by f g is the function from A to C defined by ( f g )( a ) = f ( g ( a )) . 3. Prove that for any sets A , B , and C [15] ( A B ) C = ( A C ) ( B C ) . Draw Venn diagrams of both sides of the equality. Method 1. ( A C ) ( B C ) = { a | a ( A C ) a negationslash∈ ( B C ) } = { a | a A a negationslash∈ C ∧ ¬ ( a ( B C )) } = { a | a A a negationslash∈ C ∧ ¬ ( a B a negationslash∈ C } = { a | a A a negationslash∈ C ( a negationslash∈ B a C ) } = { a | a A (( a negationslash∈ C a negationslash∈ B ) ( a negationslash∈ C a C ) } = { a | a A
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