-Pay attention to how the word “not” acts in this statement- Often you must adjust the tense of the verb or add helping verbs to make the sentence “sound” right without changing the logical structure of the sentence
Geo 2.1 condtional statements
Unit 2. Section 2.1CONDITIONAL STATEMENTS
- Conditional statements are used in every field of human endeavor.- They are crucial to the search for truth- If you are going to be able to adequately interpret and judge the statements you hear, you must understand the structure of Conditional statements.-WARNING: Once you get good at these, you may be amused by statements made by politicians and other public speakers.
Objectives Students will be able to: Identify conditional statements and place them in if-then form Identify the hypothesis and conclusion of a conditional statement Convert conditional statements into their other logical variants Identify and use truth relationships of conditional statements State and use the point, line and plane postulates of geometry
What is a ConditionalStatement? A statement that can be written in the format: If …., then …. The part after “if” is called the hypothesis Do not confuse this with the word “hypothesis” from science The rest (after “then”) is the conclusion
Statements Not in If-Then Form- Often statements are not in if –then form, but totest them out scientifically, we must convert them- In English, there are infinite ways to rephrase aconditional statement; however, we will cover thethree most common variations here
Standard Sentence Split the subject and predicate. Add “If it is” or “If they are” to the subject Add “then it” or “then they” to the predicate Smooth out the grammar Example: “All dogs go to heaven.” “If they are dogs, then they go to heaven.”
“Whenever” or “When” Replace “whenever” or “when” with “if” Add “then” after the comma Example: “Whenever I see a seagull, I think of home.” If I see a seagull, then I think of home.
“If” at the end Move the “if” clause to the beginning Add “then” after the “if” clause Example: “I eat if I am hungry.” “If I am hungry, then I eat.”
Logical Variations of theConditional-Often a conditional statement can be difficult toprove or unwieldy to use.- By using logical variations, we find forms easier toprove or use.
Converse, Inverse, &Contrapositive Converse: formed by swapping the hypothesis and the conclusion Inverse: formed by negating the hypothesis and conclusion Contrapositive: formed by both negating and swapping the hypothesis and conclusion
Equivalent Statements If the Conditional is true (or false) then so is the Contrapositive and vice versa. Similarly, if the Converse is true, then so is the Inverse and vice versa If both the Conditional and its Converse are true, then they can be rewritten as a Biconditional statement (more next class)
Example 1 If you added 2+2, you got 4 CONVERSE: If you got 4, then you added 2+2. INVERSE: If you did not add 2+2, you did not get 4. CONTRAPOSITIVE:If you did not get 4, then you did not add 2+2.
Example 2 (the word “not”) If you do not eat, you will be hungry CONVERSE: If you are hungry, then you did not eat. INVERSE: If you ate, then you are not hungry. CONTRAPOSITIVE: If you are not hungry, then you ate.
Point, Line & Plane Postulates5. Through any two points there exists exactly one line.6. A line contains at least two points.7. If two lines intersect, then their intersection is exactly one point.8. Through any three noncollinear points there exists exactly one plane.9. A plane contains at least three noncollinear points.10. If two points lie in a plane, then the line containing them lies in the plane.11. If two planes intersect, then their intersection is a line.