1_Mathematical_Preliminaries.pdf - Mathematical Preliminaries Mathematical Preliminaries • Sets • Functions • Relations • Graphs • Proof

1_Mathematical_Preliminaries.pdf - Mathematical...

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Mathematical Preliminaries
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Mathematical Preliminaries Sets Functions Relations Graphs Proof Techniques
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} 3 , 2 , 1 { A A set is a collection of elements SETS } , , , { airplane bicycle bus train B Membership A 1 B ship
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Set Representations C = { a, b, c, d, e, f, g, h, i, j, k } C = { a, b, …, k } S = { 2, 4, 6, … } finite set infinite set
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Some important sets
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Set Representations
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Set Representations Order not important S = {a, b, c, d} = {b, c, a, d}. Listing more than once does not change the set. S = {a, b, c, d} = {a, b, c, b, c, d}.
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Interval Notations
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Universal Set A universal set is the collection of all objects in a particular context or theory. Depends the context or theory under consideration. For example, we might define U as the set of all living things on planet earth. In that case, the set of all dogs is a subset of U , the set of all fish is another subset of U , and the set of all trees is yet another subset of U .
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U = { 1 , … , 10 } A = { 1, 2, 3, 4, 5 } 1 2 3 4 5 A U 6 7 8 9 10
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Set Operations A = { 1, 2, 3 } B = { 2, 3, 4, 5} Union A U B = { 1, 2, 3, 4, 5 } Intersection A B = { 2, 3 } Difference A - B = { 1 } B - A = { 4, 5 } U A B 2 3 1 4 5 2 3 1 Venn diagrams
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A Complement Universal set = {1, …, 7} A = { 1, 2, 3 } A = { 4, 5, 6, 7} 1 2 3 4 5 6 7 A A = A
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0 2 4 6 1 3 5 7 even { even integers } = { odd integers } odd
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