1. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Section 3.3 Logical Equivalence and De Morgan’s Laws

2. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Objectives o Determine the validity of formal arguments

3. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Variations on Conditional Statements Switching the order of the simple statements in conjunctions and disjunctions makes no difference to the truth value. However, this is not the case when we switch the order in conditional statements. When the order of the statements is rearranged in a conditional statement, new conditions are set forth. Given a conditional statement, we can subsequently consider the converse, inverse, and contrapositive of the original conditional statement. Note that these new variations are conditional statements themselves.

4. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Variations on Conditional Statements Table 1: Variations on Conditional Statements NameSymbolsRead Conditional p ⇒ q If p, then q Converse of Conditional q ⇒ p If q, then p Inverse of Conditional ∼ p ⇒ ∼ q If not p, then not q Contrapositive of Conditional ∼ q ⇒ ∼ p If not q, then not p Biconditional p ⇔ q p if and only if q

5. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Writing Variations of a Conditional Statement Write the converse, inverse, contrapositive, and biconditional variations of this famous conditional statement by Marilyn Monroe. a ⇒ b: If you give a girl the right shoes, then she can conquer the world. Solution Converse: b ⇒ a: If she can conquer the world, then the girl was given the right shoes. Inverse:  a ⇒  b: If you do not give a girl the right shoes, she cannot conquer the world.

6. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Writing Variations of a Conditional Statement (cont.) Contrapositive:  b ⇒  a: If she cannot conquer the world, then the girl was not given the right shoes. Biconditional: a  b: Give a girl the right shoes if and only if she can conquer the world.

7. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Writing the Contrapositive of a Conditional Statement Given the following conditional statement, write the contrapositive. If I cannot find my phone, then it is in the car. Solution Let the following represent a and b. a: I can find my phone. b: My phone is in the car.

8. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Writing the Contrapositive of a Conditional Statement (cont.) The original conditional statement is then  a ⇒ b. The contrapositive of the conditional statement  a ⇒ b is  b ⇒ a. Now we need to translate this compound statement into words. Therefore, the contrapositive to the conditional statement given is: If my phone is not in the car, then I can find it.

9. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Writing a Biconditional Statement Let the following statements represent c and d. c: I am breathing. d: I am alive. Write the biconditional statement c ⇔ d using words. Solution Biconditional statements use the phrase “if and only if.” Using statements c and d, the biconditional statement is: I am breathing if and only if I am alive.

10. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #1 Write the inverse to the statement “If I cannot find my phone, then it is in the car.” Answer: If I can find my phone, then it is not in the car.

11. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Truth Values of Variations on Conditional Statements Table 3: Truth Table ab ∼a∼a ∼ b∼ b Conditional a ⇒ b Converse b ⇒ a Inverse ∼ a ⇒ ∼ b Contrapositive ∼ b ⇒ ∼ a Biconditional a ⇔ b TTFFTTTTT TFFTFTTFF FTTFTFFTF FFTTTTTTT

12. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Biconditional Statement Biconditional statements, read “if and only if,” are true only when each component of the statement has the same truth value; that is, either both are true or both are false.

13. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Logically Equivalent Statements Logically equivalent statements are statements that have exactly the same truth values in all situations. We write this mathematically using the symbol ≡. Note in Table 3 that the conditional and its contrapositive have the same truth values. Therefore, they are logically equivalent.

14. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #2 Find the other statements that are logically equivalent in Table 3. Answer: The converse is logically equivalent to the inverse.

15. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. De Morgan’s Laws 1. ∼ ( p ∧ q ) ≡ ∼ p ∨ ∼ q 2. ∼ ( p ∨ q ) ≡ ∼ p ∧ ∼ q Augustus De Morgan, an English mathematician and logician, formally defined two famous equivalent negations that show how to negate and statements and or statements. De Morgan’s Laws show how a negative sign is “distributed” across compound statements.

16. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Applying De Morgan’s Laws Consider the following compound statement. It is not true that Jack and Jill went up the hill. From the given statements, choose the statement that is logically equivalent. a: Jack did not go up the hill and Jill did not go up the hill. b: It is not true that Jack and Jill did not go up the hill. c: Jack went up the hill or Jill went up the hill. d: Jack did not go up the hill or Jill did not go up the hill.

17. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Applying De Morgan’s Laws (cont.) Solution To determine which statement is logically equivalent, let us first write the original compound statement in symbolic form. Let the following statements represent p and q. p:Jack went up the hill. q:Jill went up the hill.

18. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Applying De Morgan’s Laws (cont.) Then our statement “It is not true that Jack and Jill went up the hill” can be written logically as ∼ ( p ∧ q ). By De Morgan’s Laws, we know that a compound statement that reads ∼ p ∨ ∼ q is logically equivalent.

19. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Applying De Morgan’s Laws (cont.) Write each of our choices symbolically as well. a: Jack did not go up the hill and Jill did not go up the hill: ∼ p ∧ ∼ q b: It is not true that Jack and Jill did not go up the hill: ∼ ( ∼ p ∧ ∼ q ) c: Jack went up the hill or Jill went up the hill: p ∨ q d: Jack did not go up the hill or Jill did not go up the hill: ∼ p ∨ ∼ q Therefore, the logically equivalent statement is d: Jack did not go up the hill or Jill did not go up the hill.

20. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #3 Determine which of the statements from Example 4 is equivalent to “Neither Jack nor Jill went up the hill.” Answer: Jack did not go up the hill and Jill did not go up the hill:  p ∧  q

21. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Negation of Conditional Statements ∼ (p ⇒ q) ≡ p ∧ ∼ q Note that the negation of a conditional statement is not itself a conditional statement.

22. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Writing the Negation of a Conditional Statement Write the negation of the following conditional statement. If I go to Moss’ Diner, then I get the triple stack pancakes. Solution First, write each piece of the conditional statement symbolically. Let the following statements represent a and b. a: I go to Moss’ Diner. b: I get the triple stack pancakes.

23. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Writing the Negation of a Conditional Statement (cont.) Our original conditional statement is then written symbolically as a ⇒ b. We know from the rule for the negation of conditional statements that the negation of a conditional statement a ⇒ b is the compound statement a ∧  b. So, we can write the negation of the conditional statement as the compound statement I go to Moss’ Diner and I do not get the triple stack pancakes.

24. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Writing the Negation of a Conditional Statement Negate the following conditional statement by using the rule of negation of conditional statements along with De Morgan’s Laws. Remember that the solution will be a compound statement. a ⇒ (c ∧ d) Solution From the negation rule we know that the negation of the conditional statement is a ∧ ∼ (c ∧ d).

25. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Writing the Negation of a Conditional Statement (cont.) We can then use De Morgan’s Laws to negate the latter half of the statement  (c ∧ d) to be ∼ c ∨ ∼ d. So, we have that the negation of a ⇒ (c ∧ d) is a ∧ ( ∼ c ∨ ∼ d).