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Mathematical Relations. Dr. Ron Smith Graceland University. Me. Father. Daughter. Widow. Mathematical relations. We can draw a picture to illustrate a relation between members of a set. Me. Father. Daughter. Widow. Mathematical Relations. Spouse. Me. Father. Daughter. Widow.

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Mathematical Relations

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Mathematical Relations

Dr. Ron Smith

Graceland University


Me

Father

Daughter

Widow

Mathematical relations

  • We can draw a picture to illustrate a relation between members of a set.


Me

Father

Daughter

Widow

Mathematical Relations

  • Spouse


Me

Father

Daughter

Widow

Mathematical Relations

  • This is a directed graph (digraph).

  • Dots are vertices, arrows are edges.


Me

Father

Daughter

Widow

Mathematical Relations

  • Birth Parent


Mathematical relations

  • A step parent is the spouse of a birth parent.

  • A parent is either a birth parent or a step parent.

  • A grandparent is a parent of a parent.

  • We combine these simpler relations to get the complex relation “Grandparent”


M F W D

Me

Father

M

F

W

D

S =

Daughter

Widow

Adjacency Matrices

  • Construct an adjacency matrix for “Spouse”.

0 0 1 0

0 0 0 1

1 0 0 0

0 1 0 0


M F W D

Me

Father

M

F

W

D

B =

Daughter

Widow

Adjacency Matrices

  • “Birth Parent”

0 0 0 0

1 0 0 0

0 0 0 1

0 0 0 0


Adjacency Matrices

  • To combine “Spouse” and “Birth Parent” to get “Step Parent” we simply multiply the adjacency matrices!

  • To combine “Birth Parent” or “Step Parent” to get “Parent” we add the adjacency matrices.


Spouse

OF

Birth Parent

0

0

1

0

0

0

0

0

0

0

0

1

0

0

0

1

1

0

0

0

0

0

0

0

Me

Father

1

0

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

0

1

0

0

0

Daughter

Widow

A step parent is the spouse of a birth parent

Step Parent!


Birth Parent

Step Parent

OR

0

0

0

0

0

0

0

1

0

0

0

1

1

0

0

0

0

0

0

0

1

0

0

0

Me

Father

0

0

0

1

0

0

0

0

0

0

0

1

0

0

0

0

1

0

0

0

1

0

0

0

Daughter

Widow

A parent is a birth parent OR a step parent

Parent!


Technical Note

  • We need to use computer logic instead of arithmetic

  • x AND y = 1 if x and y are both 1, and 0 otherwise.

  • We use • to stand for AND

1•1=

1

0•1=

0

1•0=

0

0•0=

0


Technical Note

  • x OR y is 1 if either x or y (or both) is 1, and 0 otherwise.

  • We use + to stand for OR

1+1=

1

0+1=

1

1+0=

1

0+0=

0


Technical Note

  • In fact, the logical operations AND and OR work exactly like multiplication and addition except for the fact that 1+1=1 instead of 2.

  • If we do matrix operations using AND and OR on adjacency matrices, we get the adjacency matrix of the combined relation!


Parent

OF

Parent

0

0

0

1

0

0

0

1

1

0

0

0

1

0

0

0

1

0

0

0

0

0

0

1

Me

Father

0

0

0

1

0

0

0

1

1

0

0

0

1

0

0

0

1

0

0

0

0

0

0

1

Daughter

Widow

A grandparent is a parent of a parent

Grandparent!


I’m my own Grandpa!

I’m my own Grandpa,

And my daughter’s her own Grandma,

It seems funny, I know, but I

Mathematically show-- that

I’m my own Grandpa!


The End


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