Logic/Implication and equivalence

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(Created page with 'Logic > Implication and equivalence == Contents == * Category:Mathematical logic（Wikipedia） * [[wikipedia:Boolean_algebra_(l…')
 
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* [[wikipedia:Material_conditional|Material conditional]]
* [[wikipedia:Material_conditional|Material conditional]]
* [[wikipedia:If_and_only_if|If and only if]]
* [[wikipedia:If_and_only_if|If and only if]]
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== Overview ==
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===Implication(→)===
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"α→β" is "α implies β." or "if α then β."
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0 means false, 1 means true, and ≡ means equivalent.
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Case of implication. "P→Q" ≡ "￢P∨Q" ≡ "￢(P∧￢Q)"
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{| border="1" class="wikitable" style="background-color:#ddf"
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! style="background:#ffdead;" |　P　
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! style="background:#ffdead;" |　Q　
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! style="background:#ffdead;" |　P→Q　
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! style="background:#ffdead;" |　￢P　
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! style="background:#ffdead;" |　￢Q　
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! style="background:#ffdead;" |　￢P∨Q　
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! style="background:#ffdead;" |　P∧￢Q　
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! style="background:#ffdead;" |　￢（P∧￢Q）　
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|-
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|　0
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|　0
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|　1
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|　1
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|　1
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|　1
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|　0
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|　1
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|-
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|　0
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|　1
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|　1
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|　1
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|　0
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|-
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|　0
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|-
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|-
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|}
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===Equivalence(⇔ or ≡)===
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Case of equivalence. "P⇔Q" ≡ "(P→Q)∧(Q→P)"
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{| border="1" class="wikitable" style="background-color:#ddf"
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! style="background:#ffdead;" |　P　
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! style="background:#ffdead;" |　Q　
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! style="background:#ffdead;" |　P⇔Q　
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! style="background:#ffdead;" |　P→Q　
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! style="background:#ffdead;" |　Q→P　
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! style="background:#ffdead;" |　（P→Q）∧（Q→P）　
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|-
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|　0
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|　0
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|　1
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|　1
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|　1
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|　1
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|-
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|　0
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|　1
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|　0
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|　1
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|　0
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|　0
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|-
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|-
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|}
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== CAI Exercise ==
== CAI Exercise ==

Latest revision as of 08:34, 23 May 2014

Logic > Implication and equivalence

Contents

Contents

Overview

Implication(→)

"α→β" is "α implies β." or "if α then β."

0 means false, 1 means true, and ≡ means equivalent.

Case of implication. "P→Q" ≡ "￢P∨Q" ≡ "￢(P∧￢Q)"

　P　 　Q　 　P→Q　 　￢P　 　￢Q　 　￢P∨Q　 　P∧￢Q　 　￢（P∧￢Q）　
　0 　0 　1 　1 　1 　1 　0 　1
　0 　1 　1 　1 　0 　1 　0 　1
　1 　0 　0 　0 　1 　0 　1 　0
　1 　1 　1 　0 　0 　1 　0 　1


Equivalence(⇔ or ≡)

Case of equivalence. "P⇔Q" ≡ "(P→Q)∧(Q→P)"

　P　 　Q　 　P⇔Q　 　P→Q　 　Q→P　 　（P→Q）∧（Q→P）　
　0 　0 　1 　1 　1 　1
　0 　1 　0 　1 　0 　0
　1 　0 　0 　0 　1 　0
　1 　1 　1 　1 　1 　1


CAI Exercise

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