1. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Section 3.1 Logic Statements and Their Negations

2. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Objectives o Construct statements using logic symbols

3. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Mathematical Logic Our ordinary English language is littered with opinions, sarcasm, riddles, commandments, and the list goes on. Because of this, it is often difficult to determine the validity of many of the things we hear day to day. However, there are times when we want to determine with certainty if statements are not only factually true, but also logically true. Mathematical logic provides a consistent framework in which to evaluate claims for logical truth.

4. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Statement A statement is a complete sentence that asserts a claim that is either true or false, but not both at the same time. The following sentences are examples of mathematical statements. They are represented by lowercase letters, as is the practice in mathematical logic. a: The car is blue and the cat is black b: The first even number is 2.

5. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Paradox A paradox is a sentence that contradicts itself and therefore has no single truth value. A paradox cannot be a mathematical statement.

6. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Identifying Statements Determine if the following sentences are statements. a: It is raining outside. b: Beaches are the most beautiful place to vacation. c: Today is Monday. d: Today is Monday and tomorrow is Friday. e: I lie all the time.

7. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Identifying Statements (cont.) Solution Sentence a is a statement because it can be assigned a truth value depending on the weather outside. However, sentence b is an opinion, and therefore not a statement. Sentence c is a statement since it can be either true or false, depending on the current day the statement is read. Sentence d is a statement even though it is always false. And finally, sentence e is a paradox and not a statement since it contradicts itself and therefore has no truth value.

8. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #1 Write down two statements of your own: one that is always true and one that is always false. Answer: Answers will vary.

9. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Negation The negation of a statement is the logical opposite of that statement, or its denial. Negations always have the opposite truth value of the original statement. Consider the following statement and its negation noted by the symbol ~, read as “not.” e: 5 is a prime number.(True) ~e: 5 is not a prime number. (False)

10. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Negating Statements with Quantifiers Sometimes it takes a bit more thought to negate a statement in English. This is true when the statement contains words that are quantifiers such as all, some, none, or no. The table gives us ways to negate these quantifiers. Table 1: Negating Quantifiers QuantifierNegations All areNot all are; Some are not; At least one is not Some areNone are Some are notAll are None areThere is at least one that is; Some are

11. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Negating a Statement Negate the following statements. a: Melony is wearing a red raincoat. b: The door is not closed. c: None of the tourists brought raincoats. d: I run less than Cara.

12. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Negating a Statement (cont.) Solution ~ a:Melony is not wearing a red raincoat. ~ b: The door is not not closed. However, when we negate a negation, we are back to no negation at all. So we more commonly say, “The door is closed.” ~ c: Some of the tourists brought raincoats.

13. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Negating a Statement (cont.) ~ d: I do not run less than Cara. Notice that we could also write, “I run the same as or more than Cara.” We need both parts since the opposite of “less than” is “more than or equal to.”

14. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #2 Negate the following statement. Some of the students completed their assignments. Answer:None of the students completed their assignments.

15. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Compound Statement A compound statement is composed of two or more statements joined together using connective words such as and, or, or implies. When combining two or more statements together to form a compound statement using the word and, the symbol ∧ is used between the lower case letters for the two statements, as in p ∧ q.p ∧ q. When combining two or more statements together using the connecting word or, the symbol ∨ is used, as in p ∨ q.

16. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Using Logic Symbols for Compound Statements Involving and Use the following simple statements a and b to symbolically write the given compound statement c. a: Snow is falling. b: The sun is shining. c: Snow is falling and the sun is shining.

17. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Using Logic Symbols for Compound Statements Involving and (cont.) Solution c = Snow is falling and the sun is shining. = (Snow is falling) AND (The sun is shining) = (Snow is falling) ∧ (The sun is shining) = a ∧ b

18. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Compound Statements Involving or There are two ways to think about the word or. If the meaning intended is one or the other, but not both, this is referred to as an exclusive or. If the intended meaning is that either or both of the statements can be true, this is known as the inclusive or. Note: The inclusive or is what is used in mathematical logic.

19. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Using Logic Symbols for Compound Statements Involving or Use the following statements p and q to symbolically write the given compound statement r. p: He will go to the movies tonight. q: He will stay home to give the dog a bath tonight. r : He will go to the movies tonight or he will stay home to give the dog a bath tonight.

20. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Using Logic Symbols for Compound Statements Involving or (cont.) Solution r = He will go to the movies tonight or he will stay home to give the dog a bath tonight. = (He will go to the movies tonight) OR (He will stay home to give the dog a bath tonight) = (He will go to the movies tonight) ∨ (He will stay home to give the dog a bath tonight) = p ∨ q

21. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Logic Symbols for Compound Statements Involving Implications Two statements can be joined together using the sentence structure “if a, then b.” We call this type of combination an implication because statement a implies statement b. Both “a implies b” and “if a, then b” have identical meanings in the English language. Mathematically we use a ⇒ b to symbolically represent “if a, then b.”

22. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Using Logic Symbols for Compound Statements Involving Implications Use the following statements s and t to symbolically write the given compound statement q. s: The water temperature on Saturday is below 76.2°. t: You are allowed to wear a wetsuit in the triathlon. q: If the water temperature on Saturday is below 76.2°, then you are allowed to wear a wetsuit in the triathlon.

23. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Using Logic Symbols for Compound Statements Involving Implications (cont.) Solution q= If the water temperature on Saturday is below 76.2°, then you are allowed to wear a wetsuit in the triathlon. = If (the water temperature on Saturday is below 76.2°), then (you are allowed to wear a wetsuit in the triathlon). = If (s), then (t). = s ⇒ t

24. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Summary of Logic Symbols Table 2: Logic Symbols SymbolRead ∧ And ∨ Or ~Not ⇒ Implies

25. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #3 Write the following compound statements mathematically given the simple statements a, b, and c. a: I am hungry. b: I am tired. c: I am in college. 1. I am hungry and tired. 2. I am hungry or I am in college. 3. I am tired and not in college.

26. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #3 Answer: 1. a ∧ b 2. a ∨ c 3. b ∧ ~c