Inequalities & Coded Relations Notes

Inequalities & Coded Relations

The symbols and how they chain; the two rules that decide every conclusion — a chain must be unbroken and its arrows must point the same way; why a strict and a non-strict link give a strict result; the either-or case and the exact pair that produces it; coded inequalities where symbols such as @ and # stand for relations; direct, mixed and priority-based coded questions; a short procedure to answer any pair of conclusions — with fully worked examples.

Last reviewed 22 Sept 2026 · 5 min read

The symbols

A > B (A greater), A < B, A \geq B (greater or equal), A \leq B, A = B

FormulaThe two rules

Rule 1 — the chain must be unbroken. To compare A and D you need a link at every step: $A ? B ? C ? D$ with no gap.

Rule 2 — the arrows must point the same way. $A > B > C$ gives $A > C$ . But $A > B < C$ gives nothing between A and C.

And the strictness rule: a chain of $\geq$ signs gives $\geq$ ; if even one link in the chain is strict ( $>$ ), the result is strict ( $>$ ).

So:

Chain	Conclusion
$A > B > C$	$A > C$
$A \geq B \geq C$	$A \geq C$
$A > B \geq C$	$A > C$
$A \geq B > C$	$A > C$
$A > B < C$	nothing
$A < B > C$	nothing
$A = B \geq C$	$A \geq C$

The either-or case

Two conclusions are an either-or pair when all three of these hold:

they are about the same two terms,
neither follows on its own, and
together they cover every case the chain allows.

The form that almost always appears: the chain proves $A \geq B$ but not which of the two it is. Then " $A > B$ " and " $A = B$ " each fail alone, and together they are certain — so the answer is either I or II follows.

The other legitimate pair is $A > B$ with $A \leq B$ (or $A \geq B$ with $A < B$ ), which exhausts every possibility whatever the chain says.

Coded inequalities

A table defines symbols, for example:

P @ Q means $P \geq Q$
P # Q means $P > Q$
P & Q means $P = Q$
P % Q means $P \leq Q$

Translate every symbol into its inequality first, write the whole chain, then apply the two rules. Never work in the symbols themselves.

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Inequalities & Coded Relations

The symbols

The either-or case

Coded inequalities

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