Conditional Statements Notes Discrete Mathematics F16 PDF

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Discrete Mathematics, Lamar University, Prof. Couch, Fall 2016, Lecture Notes #9...

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CONDITIONAL STATEMENTS

CONDITIONAL STATEMENTS MATH 3311 DISCRETE MATH

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CONDITIONAL STATEMENTS

9. Conditional Statements Definition 31. The statement “If p, then q.” is known as a conditional statement. The statement p is called the hypothesis, and the statement q is called the conclusion. The symbol for a conditional statement is p ⇒ q, which may also be read as “p implies q .” p T T F F

q T F T F

p ⇒ q ¬p ¬p ∨ q T F T T Figure 5.

The truth table to the above shows that the only time that the conditional statement p ⇒ q is false is when the hypothesis p is true and the conclusion q is false. The table also shows that p ⇒ q ⇔ ¬p ∨ q . 9.1. Examples. A conditional statement might be thought of as being similar to a promise, where the statement is false only in the case where the promise is broken. (1) “If I win the lottery next Wednesday, then I will buy you a stick of gum.” The only time I would break my promise is when next Wednesday rolls around, I find that I have won the lottery and I don’t buy you a stick of gum. If, when Wednesday rolls around, I find that I didn’t win the lottery, then I can either buy you a stick of gum or not as I choose; in either case, I won’t have broken my promise. (2) “If a polygon is a triangle, then the sum of the measures of its angles is 180◦ .” The hypothesis is “a polygon is a triangle,” and the conclusion is “the sum of the measures of its angles is 180◦ .” This conditional statement is true because when the hypothesis is true, then the conclusion must be true. A logically equivalent statement is “either a polygon is not a triangle or the sum of its angles is 180◦ .” In mathematics, most axioms, definitions, and theorems are stated in the “if, then” form rather than in the “either, or” form. 10. Denials of Conditional Statements We can see from the following table that the useful denial of p ⇒ q is p ∧ ¬q. Since p ⇒ q is logically equivalent to ¬p ∨ q, its useful denial is the same as for ¬p ∨ q . p T T F F

q p ⇒ q ¬(p ⇒ q) ¬q p ∧ ¬q T F T F Figure 6.

CONDITIONAL STATEMENTS

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10.1. Examples. (1) r ⇔ (p ⇒ q): If John studies for 3 hours, he will pass his math test. p: q: ¬r: (2) r ⇔ (p ⇒ q): If Toronto is the capital of New York, then Boston is the capital of Texas. p: q: ¬r: 10.2. Exercises. Write the denial of each of the following conditional statements. What is the truth value of each statement and its denial? (1) If Manhattan was built in a day, then the Sear’s Tower in Chicago is the tallest building in the world. (2) If the Sun is a star, then Mars is a star also. (3) If Sam Houston was a president of the Republic of Texas, then he was also a governor of the state of Texas. (4) If the Amazon River is in Africa, then the Nile River is in South America. (5) If “broccoli” is the name of a vegetable, then “cabbage” is the name of a fruit....