Set Theory-functions, need helpp

• Oct 26th 2006, 09:17 PM
jenjen
Set Theory-functions, need helpp

1) Let P be the set of all U.S. presidents, and let G be the set of all ordered pairs (a,b) in P X P such that b succeeded a in office. Is G the graph of a function? Explain.

2) Prove that for each set X there is a unique function from the empty set to X, regardless of whether or not X is nonempty. Also prove that there are no functions from X to the empty set if X is nonempty.
• Oct 26th 2006, 10:11 PM
Soroban
Hello, jenjen!

#1 is a trick question . . .

Quote:

1) Let $\displaystyle P$ be the set of all U.S. presidents,
and let $\displaystyle G$ be the set of all ordered pairs $\displaystyle (a,b) \in P \times P$
such that $\displaystyle b$ succeeded $\displaystyle a$ in office.
Is $\displaystyle G$ the graph of a function?

It is true that every President had a successor (well, except Dubya).

But Grover Cleveland served two nonconsecutive terms.
. . He was the 22nd President and was succeeded by Benjamin Harrison.
. . He was reelected as 24th President and was succeeded by William McKinley.
Hence, set $\displaystyle G$ contains: (Cleveland, Harrison) and (Cleveland, McKinley).

Therefore, $\displaystyle G$ is not a function.

• Oct 26th 2006, 10:23 PM
jenjen
Hey Soroban!!

Thank you so much for the quick reply!
• Oct 27th 2006, 04:02 AM
ThePerfectHacker
Quote:

Originally Posted by jenjen
]

2) Prove that for each set X there is a unique function from the empty set to X, regardless of whether or not X is nonempty. Also prove that there are no functions from X to the empty set if X is nonempty.

I do not know what definition you are using but there are no funtions between two sets if at least one is empty.

Because the Cartesian product between sets was defined for non-empty sets. Thus, a function can only between two non-empty set because it is a type of Cartesian product.