3.1 Statements and Logical Connectives 103 If–Then Statements The conditional is symbolized by → and is read “if–then.” The statement →p q is read “If p, then q.” Some books indicate that p q → may also be read “p implies q.” Implications are discussed in Section 3.3. The conditional statement consists of two parts: The part that precedes the arrow is the antecedent, and the part that follows the arrow is the consequent. Some books refer to the antecedent as the hypothesis or premise and the consequent as the conclusion. In the conditional statement →p q, the p is the antecedent and the q is the consequent. An example of a conditional statement is “If you drink your milk, then you will grow up to be healthy.” Sometimes the word then in a conditional statement is not explicitly stated. For example, the statement “If you get an A, I will buy you a car” is a conditional statement because it actually means “If you get an A, then I will buy you a car.” Example 6 Write Conditional Statements Let m t : The mulch was purchased at the garden store. : The tools were purchased at the hardware store. Write the following statements symbolically. a) If the mulch was purchased at the garden store, then the tools were not purchased at the hardware store. b) If the tools were purchased at the hardware store, then the mulch was not purchased at the garden store. c) It is false that if the mulch was not purchased at the garden store, then the tools were not purchased at the hardware store. Solution Now try Exercise 43 Example 7 Use Commas When Writing a Symbolic Statement in Words Let p q r : Jorge is enrolled in calculus. : Jorge’s major is criminal justice. : Jorge’s major is engineering. Write the following symbolic statements in words and indicate whether the statement is a negation, conjunction, disjunction, or conditional. a) q p r ( ) →∼ ∨ b) q p r ( ) → ∼ ∨ Solution The parentheses indicate where to place the commas in the sentences. a) “If Jorge’s major is criminal justice then Jorge is not enrolled in calculus, or Jorge’s major is engineering.” This statement is a disjunction because ∨ is outside the parentheses. b) “If Jorge’s major is criminal justice, then Jorge is not enrolled in calculus or Jorge’s major is engineering.” This statement is a conditional because → is outside the parentheses. 7 Now try Exercise 57 a) m t →∼ b) t m →∼ c) m t ( ) ∼ ∼ →∼ 7
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