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Mathematical Relations

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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  1. Mathematical Relations Dr. Ron Smith Graceland University

  2. Me Father Daughter Widow Mathematical relations • We can draw a picture to illustrate a relation between members of a set.

  3. Me Father Daughter Widow Mathematical Relations • Spouse

  4. Me Father Daughter Widow Mathematical Relations • This is a directed graph (digraph). • Dots are vertices, arrows are edges.

  5. Me Father Daughter Widow Mathematical Relations • Birth Parent

  6. 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”

  7. 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

  8. 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

  9. 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.

  10. 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!

  11. 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!

  12. 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

  13. 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

  14. 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!

  15. 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!

  16. 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!

  17. The End

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