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| − | '''1.1 Basic Mathematics''' | + | ==='''1.1 Basic Mathematics'''=== |
| − | + | =1.1.1 Mathematical notation= | |
• '''≈''' : approximately equal | • '''≈''' : approximately equal | ||
| − | • '''~''' : CST · | + | • '''~''' : CST · |
| − | '''Supremum and infimum vs. maximum and minimum''' | + | '''Supremum and infimum vs. maximum and minimum''' |
The concept of supremum, or least upper bound, is as follows: Let S={a[n]} , the sequence with terms a[0],a[1],\cdots over all the nonnegative integers. S has a supremum, called \sup S , if for every n , a[n]\leq\sup S (i.e. no a[n] exceeds \sup S ), and furthermore, \sup S is the least value with this property; that is, if a[n]\leq b for all n, then \sup S\leq b for all such b . This is why the supremum is also called the least upper bound, for a bound is a number which a function, sequence, or set, never exceeds. Similarly, one can define the infimum \inf S , or greatest lower bound. | The concept of supremum, or least upper bound, is as follows: Let S={a[n]} , the sequence with terms a[0],a[1],\cdots over all the nonnegative integers. S has a supremum, called \sup S , if for every n , a[n]\leq\sup S (i.e. no a[n] exceeds \sup S ), and furthermore, \sup S is the least value with this property; that is, if a[n]\leq b for all n, then \sup S\leq b for all such b . This is why the supremum is also called the least upper bound, for a bound is a number which a function, sequence, or set, never exceeds. Similarly, one can define the infimum \inf S , or greatest lower bound. | ||
Revision as of 11:12, 16 November 2010
1.1 Basic Mathematics
1.1.1 Mathematical notation
• ≈ : approximately equal
• ~ : CST ·
Supremum and infimum vs. maximum and minimum
The concept of supremum, or least upper bound, is as follows: Let S={a[n]} , the sequence with terms a[0],a[1],\cdots over all the nonnegative integers. S has a supremum, called \sup S , if for every n , a[n]\leq\sup S (i.e. no a[n] exceeds \sup S ), and furthermore, \sup S is the least value with this property; that is, if a[n]\leq b for all n, then \sup S\leq b for all such b . This is why the supremum is also called the least upper bound, for a bound is a number which a function, sequence, or set, never exceeds. Similarly, one can define the infimum \inf S , or greatest lower bound.
