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The mathematical foundations of sets, products, relations, and functions in the context of relational databases. It includes explanations of set operations such as membership, subset, equality, union, intersection, and subtraction. The document also covers the concepts of flat products and tuples, and provides exercises to test understanding. Recommended resources include the first few chapters of any discrete maths textbook.
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Lecture 12: Sets, Products, Relations, Functions
Help and exercises for the mathematically challenged. This will prepare us for some relational algebra, and normalisation, which is largely about “functional dependencies”. Much of todays material is fromRoutledge & Kegan Paul, 1965. Sets and Groups , J. A. Green, The first couple of chapters of any discrete maths textbook willhelp.
membership we write x ∈ { x , y , z } to say that x is in the set
membership we write x ∈ { x , y , z } to say that x is in the set subset if every member of A is also in B we write A ⊆ B equality two sets are equal if they have the same members
membership we write x ∈ { x , y , z } to say that x is in the set subset if every member of A is also in B we write A ⊆ B equality two sets are equal if they have the same members union we write A ∪ B for the set containing everything in A and everything in B
membership we write x ∈ { x , y , z } to say that x is in the set subset if every member of A is also in B we write A ⊆ B equality two sets are equal if they have the same members union we write A ∪ B for the set containing everything in A and everything in B intersection we write A ∩ B for the set of things that are in both A and B subtraction A − B is the elements from A that are not in B
Z = { integers} Which of the following are true? (^1 42) ⊆ Z
Z = { integers} Which of the following are true? (^1 42) ⊆ Z (^2) { 42 } ⊆ Z (^3 42) ∈ Z
Z = { integers} Which of the following are true? (^1 42) ⊆ Z (^2) { 42 } ⊆ Z (^3 42) ∈ Z (^4) { 42 } ∈ Z
Z = { integers} Which of the following are true? (^1 42) ⊆ Z (^2) { 42 } ⊆ Z (^3 42) ∈ Z (^4) { 42 } ∈ Z (^5) { 42 } ∈ {{ 42 }} (^6) {} ⊆ Z
Describe the following sets in words: A = { x ∈ Z | 2 ≤ x } B = { x ∈ Z | x ≤ 5 }
Describe the following sets in words: A = { x ∈ Z | 2 ≤ x } B = { x ∈ Z | x ≤ 5 } Find A ∩ B and A ∪ B. And A − B.
What is { 1 } × Z?
What is { 1 } × Z? What is ({ 1 } × Z) ∩ (Z × { 2 })? Does the following equation make sense?Is it true? ( A × B ) ∩ ( B × A ) = ( A ∩ B ) × ( A ∩ B )
Is ( 1 , ( 2 , 3 )) = (( 1 , 2 ), 3 )?
