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→ ~ → ~ → → ~ → → ∧ ~ → ~ 33. If I am busy every day, then I do not work in my garden. 38. If July is not a warm month, then I am busy every day and I like flowers. Logic Related Conditionals A conditional is just a compound statement in if…then… form. Thus, related conditionals are also compound statements in if…then… form. INVERSE: To find the inverse of a conditional you just negate both statements. EX: Conditional: If it is warm, then I am hot. EX: Inverse: If it is NOT warm, then I am NOT hot. Converse: To find the converse of a conditional you just switch both statements. HOWEVER, DO NOT SWITCH THE IF AND THEN!!!!!!!!!!!!!!!!!! EX: Conditional: If it is warm, then I am hot. EX: Converse: If I am hot, then it is warm. Contrapositive: To find the contrapositive of a conditional you switch AND negate both statements. EX: Conditional: If it is warm, then I am hot. EX: Contrapositive: If I am not hot, then it is not warm. REMEMBER – NEVER SWITCH THE IF AND THE THEN, JUST THE STATEMENTS!!!!! The inverse negates both parts of the conditional The converse switches the order of the statements The contrapositive switches and negates the statements in the conditional Inverse: If it does not rain, then the grass does not get wet. Converse: If the grass gets wet, then it rains. Contrapositive: If the grass does not get wet, then it does not rain Inverse: If I do not get my allowance, then I do not go to the movies. Converse: If I go to the movies, then I get my allowance Contrapositive: If I do not go to the movies, then I did not get my allowance. The inverse negates both parts of the conditional The converse switches the order of the statements The contrapositive switches and negates the statements in the conditional Inverse: If the car starts, then I will not be late for school. Converse: If I am late for school, then the car did not start. Contrapositive: If I am not late for school, then the car did start. Inverse: If I do not pass the Algebra Exam, then I will not graduate. Converse: If I graduate, then I passed the Algebra exam. Contrapositive: If I did not graduate, then I did not pass the Algebra exam. Inverse: If you did not lose the key, then you are not locked out. Converse: If you are locked out, then you lost the key. Contrapositive: If you are not locked out, then you did not lose the key. 3. : ~ → ~ 3. Converse: → 3. Contrapositive: ~ → ~ 4. : ~ → 4. Converse: ~ → 3. Contrapositive: → ~ 11. Converse: If you eat Quirky oatmeal, then you lower your cholesterol. 12. Converse: If you get rich, then you enter the Grand Prize drawing. Logically Equivalent A conditional and its corresponding contrapositive are LOGICALLY EQUIVALENT. That means that they have the same truth value! , ℎ → , ℎ ~ → ~ . Homework •Page 13 #6,7,8 •Page 14 – 15 #5,6,13,14 The inverse negates both parts of the conditional The converse switches the order of the statements The contrapositive switches and negates the statements in the conditional Inverse: If the weather is not nice, then graduation will not be outside. Converse: If graduation is outside, then the weather is nice. Contrapositive: If graduation is not outside, then the weather is not nice. Inverse: If Jack does not go up the hill, then he will not break his crown. Converse: If Jack breaks his crown, then he went up the hill. Contrapositive: If Jack does not break his crown, then he does not go up the hill. Inverse: If the temperature is above 80, then we will not go swimming. Converse: If we swimming, then the temperature is above 80. Contrapositive: If we do not go swimming, then the temperature is not above 80. 5. : → ~ 6. : → 5. Converse: → ~ 6. Converse: ~ → ~ 5. Contrapositive: ~ → 6. Contrapositive: → 13. Converse: If your hair will curl, then you use Shiny’s hair cream. 14. Converse: If your pet grows three inches, then you fed it Krazy Kibble.