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I am learning calculus/real analysis with Apostol's Calculus (2nd Edition). I have a doubt about the grammer of this book. Apostol, everywhere, uses a supremum (or a least upper bound) and an infimum (or a greatest lower bound) for some set $S$.

How many suprema and infima can a set have that he has used determinents like a and an instead of the.

As far as my learning is concerned, I was sure about the fact that supremum and infimum are unique for a given set.

Gaurav Tiwari
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    Supremum and infimum of a set are unique extended real numbers. –  Jan 17 '12 at 11:00
  • @Kannappan Exactly this is what I know. PS: Kannappan edited their comment and my comment looks regardless. – Gaurav Tiwari Jan 17 '12 at 11:03
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    Correct usage of the articles a(n) and the can be tricky. The following are correct: (1) $A$ has a supremum. (2) The supremum of $A$ is $1$. ‘$A$ has the supremum’ by itself is incorrect, and ‘A supremum of $A$ is $1$’ is somewhat misleading. – Brian M. Scott Jan 17 '12 at 11:17
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    The set ${a,a,a,a}$ has exactly one element. For every anti-symmetric relation, suprema and infima are unique. – Michael Greinecker Jan 17 '12 at 11:20
  • @Brian So clear! What if we say, '$A$ is the supremum of set $S$' at the place of ''$A$ is a supremum of set $S$''. You didn't discuss in this way! – Gaurav Tiwari Jan 17 '12 at 11:23
  • @MichaelGreinecker that was a bad example (which I've deleted) see my answer. – Seamus Jan 17 '12 at 11:24
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    ‘$a$ is the supremum of $S$’ is fine. (I used $a$ instead of $A$ because the upper-case $A$ suggests a set rather than a single real number.) – Brian M. Scott Jan 17 '12 at 11:28
  • @Brian Apostol has used $B$ for supremum of a set $S$ in the book. Probably he meant with a set, instead of a real number. – Gaurav Tiwari Jan 17 '12 at 11:32
  • possible duplicate of How many infima and suprema can there be? – Asaf Karagila Jan 17 '12 at 12:37
  • See the answers: here and here. – Asaf Karagila Jan 17 '12 at 12:38
  • To expand on Brian M. Scott's comment, saying "$A$ has the supremum." is grammatically incorrect. Saying "$A$ has the sumpremum 3" is grammatically correct. If you know that $A$ has a supremum, but you don't know its value, the indefinite article is appropriate to use (as it was in this very sentence). – David Mitra Jan 17 '12 at 12:43
  • (Not that it's very relevant for the question, but the correct spelling is "grammar", not "grammer".) – Hans Lundmark Jan 17 '12 at 14:20
  • (And also not directly relevant to the question, but the plural of "supremum" is "suprema", and the plural of "infimum" is "infima"; then again, the plural of "stadium" is "stadia", and that's not commonly used outside of Tom Lehrer albums...) – Arturo Magidin Jan 17 '12 at 16:17

2 Answers2

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Let $(X,\leq)$ be a partially ordered set. That means that for all $x,y,z\in X$:

1) $x\leq x$ (reflexivity)

2) $x\leq y$ and $y\leq x$ imply $x=y$ (anti-symmetry)

3) $x\leq y$ and $y\leq z$ imply $x\leq z$ (transitivity)

An element $x\in X$ is a lower bound of a subset $S\subseteq X$ if $x\leq S$ for all $s\in S$. An element $x$ is the greatest element in a subset $T\subseteq X$ if $x\in T$ and $t\leq x$ for all $t\in T$. An infimum of a set $S$ is a greatest element in the set of lower bounds of $S$. We will show that there can be at most one greatest element in every set, so there can be at most one infimum for every set.

There can be at most one greatest element in a subset $T\subseteq X$.

Proof: Let $x,x'$ be both greatest elements in $T$. Then $x\in T$ and $x'\in T$ and since $x$ is a greatest element in $T$, we have $x'\leq x$. Similarly, $x\leq x'$. By anti-symmetry, $x=x'$.

Michael Greinecker
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In general, you can have more than one infimum. Consider sets of pairs of numbers $(a,b)$ and the relation on pairs $(a,b) \succeq (c,d)$ iff $a\ge c$ and $b\ge d$. This is only a partial order, and so, with respect to this relation, the following set has two distinct infima: $\{(1,2),(2,1),(2,2)\}$. That is, there are two elements that satisfy: "no element of the set is less that this element".

If the set is totally ordered (like the reals) then the infimum and supremum are unique.

Seamus
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    What you call suprema and infima are usually known as maximal and minimal elements. The common definition of an infimum is "the largest lower bound" and a supremum is the "smallest upper bound". In particular, an infimum is comparable with every other lower bound (similar for suprema), which means, by anti-symmetry of every partial order, that an infimum, if it exists, is unique. – Michael Greinecker Jan 17 '12 at 11:29
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    This is incorrect. If $\langle P,\preceq\rangle$ is a partial order, the infimum $m$ of a set $S\subseteq P$ must satisfy two conditions: $m\preceq s$ for all $s\in S$, and if $t\preceq s$ for all $s\in S$, then $t\preceq m$. In other words, it must be the greatest lower bound. Your partial order has no infimum, though it has two minimal (not minimum) elements. – Brian M. Scott Jan 17 '12 at 11:32
  • Sometimes it is very odd to say 'a largest' and 'a smallest'. 'the' is written before superlative degree of adjectives. – Gaurav Tiwari Jan 17 '12 at 11:36
  • I think I was taught that the first condition for $s$ being a supremum was $\neg (m \succ s)$. But perhaps I'm wrong. Anyway, the answer is sensitive to the definition of supremum. – Seamus Jan 17 '12 at 12:16