WARM-UP. Conditional Statements

Transcription

1 WARM-UP B A 4x + 33 C D 2x +81 ( ) + 33 ind the values of the three angles below m ABD 4x + = 33 4= 8 2x +81 m ABD +4x = 32 +4x + 33= = 6x + 81 m DBC 81 = 2( 8) m DBC 48 = 16 = 6x +81= m ABC 8 = 2= ( 65 x )=130 You need to begin by Now finding for the the angles! value of x Conditional Statements SECION 2.1 1

2 oday s opics Conditional Statements Definition & Shortcut Notations Re-writing a statement as an if-then Applying Negation to statements Related Statements Inverse Converse Contra-Positive Shorthand Symbols for Perpendicular & Parallel Biconditional Statements ruth ables Conditional Statement A logical statement that begins with a hypothesis you are in Portland that is followed you by are a conclusion. in Oregon hese statements are often referred hypothesis to as if-then statements conclusion Often times in mathematics the logic (not If words) you are are in more Portland, important thus then you are a in shorthand Oregon. p q p: he hypothesis & conclusion are q: broken up by the then p implies q Let s take a look at an example of an if-then statement 2

3 Exploration Write down three different if-then statements. Statement #1: A rue Statement Statement #2: A alse Statement Statement #3: One that is difficult to decide! Once If a finished month has please 30 days, share then them it with is ebruary. the person sitting next to you (your elbow partner). If See an if angle he (or is she) right, can then spot it has which a measure one is which!! of 90. No! Sept., Apr., Jun., & Nov. Yes! his is the definition. You If you have are 15 in minutes! Eugene, Be then prepared you are to in share Oregon. out to the class!! No! Indiana & Missouri. Re-writing a Statement You will be asked to re-write conditional statements that lack an if & a then as if-then statements All kitty cats have claws. If it s a cat, then it has claws. 4x + 5 = 13, if x = 2. If x = 2, then 4x + 5 = 13 3

4 Re-writing a Statement Your turn! Re-write the following conditional statements using if & then oday is riday, and tomorrow is the weekend! 2x + 7 = 1, because x = -3. Stop Being So Negative Often here times is a short we will hand want for the negation opposite just of like the there is for hypothesis hypothesis or conclusion. and the conclusion not p ~ p p his is achieved by using negation! Original Statements he balloon is red. he sky is not dark. Negated Statements he balloon is not red. he sky is dark. 4

5 Related Conditionals We achieve three more conditional statements based on the original if-then using order & negation Original Statement If it is raining outside, then you will be wet! Converse: Contra-Positive: Inverse: Re-ordering Negating Hypothesis Both When Hyp. hypothesis & conclusion Con. are & negated conclusion are negated & switched If If you it you is not are wet, raining wet, then then outside, it it is is not then raining you outside! will not get wet! Equivalent Statements hese Now If we decide: related organize if conditionals the these hypothesis into match a table was up we true, in see terms would the of their value the conclusion connections! or the logic be they true. express. If the stars are visible, then it is night. OR. Words Symbol Value rue Please Note: If Original it is night, then p q the stars are Write visible. Sometimes Equivalent CON. the following alse all statements will Converse q p conditional If the stars are not visible, then statement it is not be true, night. and IN. in the alse Inverse ~ p ~ q other times form of it s inverse, If it is not night, then the stars are converse, not visible. C.P. Contra-P. ~ q ~ p they Equivalent will all be & rue false. contra-positive. 5

6 Parallel & Perpendicular wo Did lines you notice are parallel the link between wo lines the are hypothesis perpendicular & conclusion if and only in these if definitions? if (If and and only only if if) they do not intersect. they form a right angle. m n m n linem / / linen linem linen Biconditional Statements A statement BU what that about represents a shortcut the original for these and converse statements?! in one statement Original p q q p p iff q p q If a shape is a triangle, then it has three sides. Converse hese types of statements are popular both If a because shape A shape has they is three a can triangle sides, be read iff then it both has it is forwards three a triangle. sides. & backwards, but often times definitions in math Biconditional are written using biconditional statements. A shape is a triangle if and only if it has three sides. 6

7 ruth ables Handy little device we use to compare the value of conditional statements & all of the possible scenarios. Let s consider the following p Which Let s q two seem take statements: reasonable? his a look is the at each truth I am of these in Paris, options I table am in from for rance the the Could both of these be lens of the conditional original true? Both alse? I am in conditional Paris, p statement. : I am I am not stance. in in Paris rance OR Could one or the other I am When not in the q Paris, : ALL hypothesis is I am I am original in in rance be true? Let s organize true what do conditional you expect these choices by a table the conclusion statements!! to be? I am not in Paris, I am not in rance ruth ables Any easy trick for these is to remember that the ry your only hand time at a truth statement tables. is Please false is complete the following if it starts table true, for the but converse ends false & inverse. p q ~p ~q q p ~p ~q 7

8 Homework!! Perfect!! Pg #10-12, 14-15, 17-18, 34-35, 39-40, 47,