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Aim 1.7: To explore the law of contrapositive, demorgan’s law & Chain rule

Do Now: - Write the Contrapositive truth table Write the contrapositive of the following statement: “If I go to school, then I will take a test” Homework: Packet p. 11 #2 – 5, #7 & 8. Aim 1.7: To explore the law of contrapositive, demorgan’s law & Chain rule. Given:

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Aim 1.7: To explore the law of contrapositive, demorgan’s law & Chain rule

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  1. Do Now: - Write the Contrapositive truth table Write the contrapositive of the following statement: “If I go to school, then I will take a test” Homework: Packet p. 11 #2 – 5, #7 & 8 Aim 1.7: To explore the law of contrapositive, demorgan’s law & Chain rule

  2. Given: If the sky is blue, then I will get ice cream. The sun is yellow The sky is blue or it is green. If the sun is yellow, then the sky is not green. Practice: Aim 1.7: To explore the law of contrapositive, demorgan’s law & Chain rule

  3. Statement Reason The Sun is yellow 1. Given If the sun is yellow, then 2. Given the sky is not green. 3. The sky is not green. 3. Law of Detachment 1,2 4. The sky is blue or green. 4. Given 5. The sky is blue. 5. LODI 3, 4 6. If the sky is blue, then 6. Given we will get ice cream. 7. We will get ice cream. 7. Law of Detachment 5, 6 Practice: Aim 1.7: To explore the law of contrapositive, demorgan’s law & Chain rule

  4. Given: a p V t a  ~t Prove: p Statement Reason 1. a 1. Given 2. a  ~t 2. Given 3. Therefore ~t 3. Law of Detachment 1,2 4. p V t 4. Given 5. Therefore p 5. LODI 3, 4 Example Aim 1.7: To explore the law of contrapositive, demorgan’s law & Chain rule

  5. The conditional and it’s contrapositive are logically equivalent (same truth value). • Either they’re both true or both false. p  q If p implies q is true ~q  ~p therefore, not q implies not p is also true Example: If I do not practice, then I will not win. Contrapositive: If I win, then I practiced. Law of the Contrapositive Aim 1.7: To explore the law of contrapositive, demorgan’s law & Chain rule

  6. Given: x V y ~ x z  ~ y Prove: ~ z Statement Reason 1. x V y 1. Given 2. ~ x 2. Given 3. y 3. LODI 1, 2 4. z  ~ y 4. Given 5. y  ~z 5. Law of Contrapositive 6. ~ z 6. Law of Detachment 3, 5 Example: Aim 1.7: To explore the law of contrapositive, demorgan’s law & Chain rule

  7. Given: a  b ~c V ~ b c Prove: ~a Statement Reason c 1. Given ~c V ~b 2. Given Therefore ~b 3. LODI 1, 2 a b 4. Given ~b  ~a 5. Law of Contrapositive ~a 6. Law of Detachment 3, 5 Example Aim 1.7: To explore the law of contrapositive, demorgan’s law & Chain rule

  8. DeMorgan’s: If we negate a conjunction or a disjoint sentence, then we must negate each statement and change conjunction to a disjunction, or vice versa De Morgan's laws creates a logically equivalent statement ~ ( p V q )  ~p ^ ~q ~ ( p ^ q )  ~p V ~q DeMorgan’s law Aim 1.7: To explore the law of contrapositive, demorgan’s law & Chain rule

  9. Write the negation of the following sentences: • ~ ( x V y ) ~x ^ ~y • ~ a V ( t ^ ~ x) a ^ ( ~t V x ) 3. ~ ( q V p ) ^ ( w v r ) ( ~ q ^ ~ p ) V ( ~ w ^ ~ r ) Demorgan’s law Aim 1.7: To explore the law of contrapositive, demorgan’s law & Chain rule

  10. Given: ~ ( p ^ q ) ~ q  t p Prove: t Statement Reason ~ ( p ^ q ) 1. Given ~ p V ~ q 2. DeMorgan’s Law p 3. Given ~ q 4. LODI 2, 3 ~ q  t 5. Given t 6. Law of Detachment 4, 5 DEMOrgan’s Law Aim 1.7: To explore the law of contrapositive, demorgan’s law & Chain rule

  11. The Chain Rules allows us to make a logical conclusion using two Conditional statements: If today is Monday, then I have science lab. If I have science lab, then I have two periods of science. Chain Rule: If today is Monday, then I have two periods of science a  b b  c Chain Rule: a  c CHAIN Rule Double Detachment Aim 1.7: To explore the law of contrapositive, demorgan’s law & Chain rule8

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