- Understanda biconditionalstatement.
Introduction
In geometrywe reasonfrom knownfacts and relationshipsto createnew ones.You saw earlierthat inductive
reasoningcan help,but it doesnotproveanything.For that we needanotherkind of reasoning.Now you
will beginto learnaboutdeductivereasoning,the kind of reasoningusedthroughoutmathematicsand
science.
If-ThenStatements
In geometry, and in ordinarylife, we oftenmakeconditional, orif-then, statements.
- Statement1: If the weatheris nice,I’ll washthe car. (“Then”is impliedevenif not stated.)
- Statement2: If you workovertime,then you’llbe paid time-and-a-half.
- Statement3: If dividesevenlyinto , then is an evennumber.
- Statement4: If a trianglehas threecongruentsides,it is an equilateraltriangle.(“Then”is implied;this
is a definition.) - Statement5: Allequiangulartrianglesareequilateral. (“If” and “then”are both implied.)
An if-thenstatementhas two parts.
- The “if” part is calledthehypothesis.
- The “then”part is calledtheconclusion.
For example,in statement2 above,the hypothesisis “you workovertime.”The conclusionis “you’llbe paid
time-and-a-half.”
Lookat statement1 above.Eventhoughthe word“then”is not actuallypresent,the statementcouldbe
rewrittenas: If the weatheris nice,thenI’ll washthe car. This is the meaningof statement1. The hypothesis
is “the weatheris nice.”The conclusionis “I’ll washthe car.”
Statement5 is a little morecomplicated.“If” and “then”are both impliedwithoutbeingstated.Statement 5
can be rewrittenas: If a triangleisequiangular,then it isequilateral.
Whatis meantby an if-thenstatement?Supposeyour friendmakesthe statementin statement2 above,
and addsanotherfact.
- If you workovertime,then you’llbe paid time-and-a-half.
- You workedovertimethis week.
If we acceptthesestatements,whatotherfactmustbe true?Combiningthesetwo statements,we can state
with no doubt:
You’ll be paid time-and-a-halfthis week.Let’s analyzestatement1, whichwas rewrittenas: If the weatheris nice,then I’ll washthe car. Supposewe
acceptstatement1 and anotherfact: I’ll washthe car.
Can we concludeanythingfurtherfrom thesetwo statements?No. Evenif the weatherisnotnice,I might
washthe car. Wedoknowthat if the weatheris nice I’ll washthe car. Wedon’tknowwhetheror not I might