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Logic Unit 1 Sentences, Statements and Truth Values Geometry Logic - the science of reasoning Mathematical sentences declarative statements that may be either true or false (has a truth value). The Ravena Library is on Main Street. mathematical sentence 17 + 5 = 21 mathematical sentence Go to your room. Not a mathematical sentence Do you have soccer practice today? Not a mathematical sentence 14 – 8 = 6 mathematical sentence Every parallelogram Not a mathematical sentence (it’s a phrase) Open Sentences - sentences that contain a variable What is the variable of each sentence? x + 2 = 8 the variable is x He broke my pencil. the variable is he Jenny found it behind the sofa. the variable is it Domain - the set of all elements that are possible replacements for the variable. Solution set (truth set) - the elements from the domain that make an open sentence true. Open sentence: It is the third month of the year. Variable: It Domain: {all the names of the months} Solution (truth) set: {March} Closed Sentences - a statement that can’t be judged as true or false. There are no variables. Negations The negation of a statement always has the opposite truth value of the original statement. Insert or take out the word ‘not’ to obtain the negation of a statement. For example: Statement: Neil Armstrong walked on the moon. True Negation: Neil Armstrong did not walk on the moon. False For example: Statement: A duck is not a mammal. True Negation: A duck is a mammal. False Tell whether the following is a mathematical sentence or not. 1. Thanksgiving is on the fourth Thursday in November. Mathematical Sentence 2. Where did you go? Not a Mathematical Sentence 3. Be quiet. Not a Mathematical Sentence 4. y – 7 = 3y + 4 Mathematical Sentence 5. The first president of the United States was Abraham Lincoln. Mathematical Sentence What is the variable in the following open sentences. 5. She is tall. 6. 2y > 17 7. This country has the third largest population. 8. It is my favorite food. Tell whether each sentence is true, false, or open. 9. The Statue of Liberty was given to the United States by France. True 10. Tallahassee is a city in Montana. False 11. 6x + 4 = 16 Open Variable: x 12. 6(2) + 4 = 16 True Find the truth set for each open sentence using the replacement set {Nevada, Illinois, Massachusetts, Alaska, New York}. 13. Its capital is Albany. {New York} 14. It is on the east coast of the United States. {Massachusetts, New York} 16. It is one of the last two states admitted to the United States. {Alaska} Write the negation of each sentence. 20. A stop sign is painted green. F A stop sign is not painted green. T 21. There are 1,760 yards in a mile. There are not 1,760 yards in a mile. 22. 14(2) – 16 = 12 14(2) – 16 ≠ 12 T F 23. Today is not Thursday. Today is Thursday. F T T F Homework Worksheet Sentences, Statements, and Truth Values 1.01