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Definitions for the terms 'bounded from above', 'bounded from below', 'bounded', 'upper bound', 'lower bound', 'least upper bound', and 'greatest lower bound' in the context of set theory. It explains what each term means and how they relate to one another.
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TERM 1
DEFINITION 1 A set S R is bounded from above if there is a real number M such that for all xS, xM. TERM 2
DEFINITION 2 A set SR is bounded frombelow if there is a real number M such that for all xS , Mx. TERM 3
DEFINITION 3 A set is bounded if it is bounded both from above and below. TERM 4
DEFINITION 4 If SR a real number M is an upper bound for S if for all xS , xM. TERM 5
DEFINITION 5 If SR a real number M is alower bound for S if for all xS , Mx.
TERM 6
DEFINITION 6 A real number a is a least upper bound for S if a is an upper bound for S with the property that if b is also an upper bound for S then ab. TERM 7
DEFINITION 7 A real number a is agreatest lower boundfor S if a is a lower bound for S with the property that if b is any otherlower bound for S then ba.