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Aim 1.8: To explore the law of contrapositive, demorgan’s law & Chain rule

Do Now: Write the contrapositive of the following statement: “If today is Wednesday, then it is an A day” Homework: Packet p. 11 #7, 8 & 11. Aim 1.8: To explore the law of contrapositive, demorgan’s law & Chain rule. Given: a  b a z  ~ b Prove: ~ z. Example:.

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Aim 1.8: To explore the law of contrapositive, demorgan’s law & Chain rule

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  1. Do Now: Write the contrapositive of the following statement: “If today is Wednesday, then it is an A day” Homework: Packet p. 11 #7, 8 & 11 Aim 1.8: To explore the law of contrapositive, demorgan’s law & Chain rule

  2. Given: a  b a z  ~ b Prove: ~ z Example: Aim 1.8: To explore the law of contrapositive, demorgan’s law & Chain rule

  3. 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 1.8: To explore the law of contrapositive, demorgan’s law & Chain rule

  4. 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 1.8: To explore the law of contrapositive, demorgan’s law & Chain rule

  5. 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 1.8: To explore the law of contrapositive, demorgan’s law & Chain rule

  6. 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 1.8: To explore the law of contrapositive, demorgan’s law & Chain rule

  7. 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 1.8: To explore the law of contrapositive, demorgan’s law & Chain rule

  8. 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 1.8: To explore the law of contrapositive, demorgan’s law & Chain rule

  9. 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 1.8: To explore the law of contrapositive, demorgan’s law & Chain rule8

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