Write each of these statements in the form “if p, then q” in...

Write each of these statements in the form “if p, then q” in English. [Hint: Refer to the list of common ways to express conditional statements provided in this section.] a) I will remember to send you the address only if you send me an e-mail message. b) To be a citizen of this country, it is sufficient that you were born in the United States. c) If you keep your textbook, it will be a useful reference in your future courses. d) The Red Wings will win the Stanley Cup if their goalie plays well. e) That you get the job implies that you had the best credentials. f) The beach erodes whenever there is a storm. g) It is necessary to have a valid password to log on to the server. h) You will reach the summit unless you begin your climb too late. i) You will get a free ice cream cone, provided that you are among the first 100 customers tomorrow.

Write each of these statements in the form “if p, then q” in English. [Hint: Refer to the list of common ways to express conditional statements provided in this section.]
a) I will remember to send you the address only if you send me an e-mail message.
b) To be a citizen of this country, it is sufficient that you were born in the United States.
c) If you keep your textbook, it will be a useful reference in your future courses.
d) The Red Wings will win the Stanley Cup if their goalie plays well.
e) That you get the job implies that you had the best credentials.
f) The beach erodes whenever there is a storm.
g) It is necessary to have a valid password to log on to the server.
h) You will reach the summit unless you begin your climb too late.
i) You will get a free ice cream cone, provided that you are among the first 100 customers tomorrow.

Key Concepts

Interpreting Alternative Conditional Phrases

Translating statements that use varied phrasing, such as 'only if', 'unless', 'provided that', 'implies', and 'whenever', into the standard 'if p, then q' form requires recognizing the underlying logical structure. Each phrasing has specific implications: for example, 'only if' often indicates a necessary condition, while 'if' signals sufficiency. Familiarity with these phrases is crucial in accurately rewriting and interpreting conditional statements across different contexts.

Conditional Statement

A conditional statement is a logical expression of the form “if p, then q,” where p is called the antecedent and q is the consequent. It establishes a relationship in which the truth of the antecedent guarantees the truth of the consequent, and it is a fundamental component in deductive reasoning and formal logic.

Necessary Condition

A necessary condition specifies that a particular condition must be met for a statement to be true. In other words, if q is true, then p must also be true when p is stated as a necessary condition. Recognizing necessary conditions is vital because they often appear in logical expressions phrased as 'only if' or 'it is necessary to,' indicating that without the antecedent, the consequent cannot hold.

Sufficient Condition

A sufficient condition indicates that the presence of one condition is enough to ensure that another condition holds true. In conditional statements, a sufficient condition is often expressed by phrases like 'if' or 'provided that,' meaning that when the antecedent is true, the consequent will be guaranteed. Understanding sufficient conditions helps in deconstructing assertions where fulfilling the antecedent assures the truth of the consequent.

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