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Mathematical Relations

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Mathematical Relations

Dr. Ron Smith

Graceland University

Me

Father

Daughter

Widow

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

Me

Father

Daughter

Widow

- Spouse

Me

Father

Daughter

Widow

- This is a directed graph (digraph).
- Dots are vertices, arrows are edges.

Me

Father

Daughter

Widow

- Birth Parent

- 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

- 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

- “Birth Parent”

0 0 0 0

1 0 0 0

0 0 0 1

0 0 0 0

- 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

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

Parent!

- 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

- 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

- 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

Grandparent!

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