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Revision as of 03:35, 31 January 2009
, 03:35, 31 January 2009→Definition 3
===Definition 3===
===Definition 3===
<pre>
<br>
If f, g : U -> V,
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px; text-align:center"
| width="80%" |
| width="20%" | <math>\text{Definition 3}\!</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px"
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="80%" style="border-top:1px solid black" | <math>f, g ~:~ X \to Y</math>
|- style="height:40px"
|
| <math>\text{then}\!</math>
| <math>\text{the following are equivalent:}\!</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px"
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D3a.}</math>
| width="60%" style="border-top:1px solid black" | <math>f ~=~ g</math>
| width="20%" style="border-top:1px solid black" |
|- style="height:40px"
|
| <math>\operatorname{D3b.}</math>
| <math>f(x) ~=~ g(x)</math>
| align="center" | <math>\forall x \in X</math>
|}
|}
<br>
<br>
D3b. f(u) = g(u), for all u C U.
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</pre>
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px; text-align:center"
| width="80%" |
| width="20%" | <math>\text{Definition 3}\!</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px"
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="80%" style="border-top:1px solid black" | <math>f, g ~:~ X \to Y</math>
|- style="height:40px"
|
| <math>\text{then}\!</math>
| <math>\text{the following are equivalent:}\!</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px"
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D3a.}</math>
| width="80%" style="border-top:1px solid black" | <math>f ~=~ g</math>
|- style="height:60px"
|
| <math>\operatorname{D3b.}</math>
| <math>\overset{X}{\underset{x}{\forall}}~ (f(x) = g(x))</math>
|}
|}
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px; text-align:center"
| width="80%" |
| width="20%" | <math>\text{Definition 3}\!</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px"
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="80%" style="border-top:1px solid black" | <math>f, g ~:~ X \to Y</math>
|- style="height:40px"
|
| <math>\text{then}\!</math>
| <math>\text{the following are equivalent:}\!</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px"
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D3a.}</math>
| width="80%" style="border-top:1px solid black" | <math>f ~=~ g</math>
|- style="height:60px"
|
| <math>\operatorname{D3b.}</math>
| <math>\prod_x^X (f(x) = g(x))</math>
|}
|}
<br>
===Definition 4===
===Definition 4===
