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# 1st Test - If then, converse, inverse and contrapositive

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### 1st Test - If then, converse, inverse and contrapositive

1. 1. Conditionals =) Conditionals make me happy If we are doing conditionals then I am happy
2. 2. Converting from… statement to conditional <ul><li>A conditional is just another way to write a statement using the words “if” and “then” </li></ul><ul><li>Our statement will be something like “if p then q” </li></ul><ul><li>It is </li></ul>
3. 3. An example <ul><li>Math teachers love their jobs </li></ul><ul><li>Let p = “Teaching Math” </li></ul><ul><li>Let q = “loving your job” </li></ul><ul><li>We are using the form if p then q </li></ul><ul><li>So…. </li></ul><ul><li>… .. </li></ul><ul><li>… . =) </li></ul><ul><li>If you teach math then you love your job </li></ul>
4. 4. Another example <ul><li>Students hate homework </li></ul><ul><li>Let p = “Being a student” </li></ul><ul><li>Let q = “hating homework” </li></ul><ul><li>We are using the form if p then q </li></ul><ul><li>If you are a student then you hate homework </li></ul>
5. 5. One More Example <ul><li>Ben’s roommate plays guitar </li></ul><ul><li>Let p = “being Ben’s roommate” </li></ul><ul><li>Let q = “playing guitar” </li></ul><ul><li>We are using the form if p then q </li></ul><ul><li>If you share rent with Ben then you play guitar </li></ul>
6. 6. The converse <ul><li>Converse just means to flip our argument around </li></ul><ul><li>Start with “if p then q” then the converse is “if q then p” </li></ul>
7. 7. Converse example <ul><li>If you teach math then you love your job </li></ul><ul><li>So p is “teaching math” </li></ul><ul><li>And q is “loving your job” </li></ul><ul><li>Since the converse is “if q then p” this specific converse is …. </li></ul><ul><li>…… </li></ul><ul><li>…… . =) </li></ul><ul><li>If you love your job then you teach math </li></ul>
8. 8. Another converse example <ul><li>If you share rent with Ben then you play guitar </li></ul><ul><li>so p is “being Ben’s roommate” </li></ul><ul><li>And q is “playing guitar” </li></ul><ul><li>The converse of the above statement would be “If you play guitar then you share rent with Ben” </li></ul>
9. 10. The inverse <ul><li>The inverse means to take the NOT of both statements </li></ul><ul><li>If we start with “If p then q” then the inverse is “if !p then !q” </li></ul><ul><li>The above reads as “If not p then not q” </li></ul>
10. 11. Inverse example <ul><li>If you are a student then you hate homework </li></ul><ul><li>So p is “being a student” </li></ul><ul><li>And q is “hating homework” </li></ul><ul><li>The inverse would be “if !p then !q” so it would be </li></ul><ul><li>“ If you are not a student then you love homework” </li></ul>
11. 12. Another inverse example <ul><li>If you teach math then you love your job </li></ul><ul><li>So p is “teaching math” </li></ul><ul><li>And q is “loving your job” </li></ul><ul><li>The inverse would be “if !p then !q” so it would be </li></ul><ul><li>“ If you do not teach math then you hate your job” </li></ul>
12. 13. The contrapositive <ul><li>The contrapositive is both the converse and the inverse at the SAME TIME </li></ul><ul><li>The collision of the two!!!!111eleventy </li></ul>
13. 14. The contrapositive <ul><li>Our starting statement is “if p then q” </li></ul><ul><li>To find the contrapositive we find the converse “if q then p” </li></ul><ul><li>Then we find the inverse of the converse “if !q then !p” </li></ul>
14. 15. Contrapositive example <ul><li>If you are a student then you hate homework </li></ul><ul><li>So p is “being a student” </li></ul><ul><li>And q is “hating homework” </li></ul><ul><li>The converse would be “If you hate homework then you are a student” </li></ul><ul><li>Then we would take the inverse of that and get “if you love homework then you are not a student” </li></ul>
15. 16. Another contrapositive example <ul><li>If you teach math then you love your job </li></ul><ul><li>So p is “teaching math” </li></ul><ul><li>And q is “loving your job” </li></ul><ul><li>The contrapositive would be “if !q then !p” </li></ul><ul><li>So we have “if you hate your job then you do not teach math” </li></ul>
16. 17. To sum it up <ul><li>Conditional– If p then q </li></ul><ul><li>Converse – if q then p </li></ul><ul><li>Inverse – If !p then !q </li></ul><ul><li>Contrapositive – If !q then !p </li></ul>
17. 18. Summing it up using geometry <ul><li>Statement: A triangle is a polygon </li></ul><ul><li>Conditional: If it is a triangle then it is a polygon </li></ul><ul><li>converse: if it is a polygon then it is a triangle </li></ul><ul><li>inverse: if it is not a triangle then it is not a polygon </li></ul><ul><li>contrapositive: if it is not a polygon then it is not a triangle </li></ul>