Relations and Functions

We know that if set A contains m elements and set B contains n elements, then the set A B will have m . n ordered pairs
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If set A contains m elements and set B contains n elements, then the set A x B will have

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Any subset of ordered pairs in A x B is called a relation from

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A x B

Let A = {1, 2, 3} and B = {2, 4}, then A x B =

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Function from A to B

A collection of ordered pairs (from the set A x B) constitute a special relation from A to B, which is called a function from A to B if we select the ordered pairs in such a way that:

their first elements constitute the entire set A
no two distinct pairs have the same first element
their second elements constitute the entire set B
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Relations from A to B

Let A = {1, 2, 3, 4} and B = {2, 4, 5}

The following are all possible relations from A to B

R1 = {(1, 2), (1, 5), (2, 2), (3, 4), (3, 5), (4, 5)}
R2 = {(1, 4), (4, 2), (4, 5)}
R3 = {(3, 2), (3, 4), (3, 5), (1, 4)}
R4 = {(1, 4), (2, 5), (3, 2), (4, 4)}
R5 = {(1, 2), (2, 5), (3, 4), (4, 4)}

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Product with empty set

Let A = {1, 2, 3}, B = ∅, Then A x B = 

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Domain of a Relation

In any relation (in the form of a set of ordered pairs), the set consisting of the ________ element of each pair constitutes the domain of the relation.

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Given a domain of R, tell if it is a function

Let A = {1, 2, 3, 4} and B = {2, 4, 5}

The following are all possible relations from A to B

R1 = {(1, 2), (1, 5), (2, 2), (3, 4), (3, 5), (4, 5)}
R2 = {(1, 4), (4, 2), (4, 5)}
R3 = {(3, 2), (3, 4), (3, 5), (1, 4)}
R4 = {(1, 4), (2, 5), (3, 2), (4, 4)}
R5 = {(1, 2), (2, 5), (3, 4), (4, 4)}

Which of these  relations represent a function

  • R1 (1) transparent a function
  • R2  (2) transparent a function
  • R3 (3) transparent a function
  • R4  (4) transparent a function
  • R5  (5) transparent a function
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represents
does not represent
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