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 May 2nd, 2015, 02:14 AM #1 Member   Joined: Jan 2010 Posts: 44 Thanks: 0 monotonic laws for ordinal subtraction I have to prove some monotonic laws for ordinals. It's quite comfortable for me to show monotonic laws of ordinal addition (e.g. $\beta\leq\gamma\Rightarrow\alpha+\beta\leq\alpha+ \gamma$). But when it comes to laws with subtraction, then I'm not sure where to start. Maybe it's because of definition of subtraction for ordinals $\alpha-\beta=\gamma$, if $\alpha=\beta+\gamma$, which is not constructive. So, maybe someone can give me a hint on how to prove those: $\alpha,\beta,\gamma$ - ordinals. (i) $\alpha>\beta\Rightarrow \gamma(\alpha-\beta)=\gamma\alpha-\gamma\beta$ (ii)$\alpha>\beta>\gamma \Rightarrow \alpha-\gamma>\beta-\gamma$ May 2nd, 2015, 05:40 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,697 Thanks: 2681 Math Focus: Mainly analysis and algebra Well you can always use $p \gt q \implies p \lt (-q)$ and $p - q = p + (-q)$. May 4th, 2015, 10:17 AM #3 Member   Joined: Jan 2010 Posts: 44 Thanks: 0 I haven't seen $\displaystyle -q$ defined for an ordinal $\displaystyle q$... Tags laws, monotonic, ordinal, subtraction Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post ZardoZ Calculus 3 August 31st, 2011 09:36 AM SysInv Advanced Statistics 1 October 25th, 2010 03:41 PM xianghu21 Applied Math 4 April 8th, 2010 01:35 PM riemann Applied Math 1 November 27th, 2008 04:00 PM thaithuan_GC Algebra 0 September 27th, 2008 02:31 AM

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