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Elementary Foundations: An introduction to topics in discrete mathematics

Jeremy Sylvestre

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Activities 1.5 Activities
A4US

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Activity 1.1.

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Consider the following statement.
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If the game is on and the popcorn is ready, then Joe is happy.
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(a)

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Assign statement variables and rewrite the statement in symbolic language.
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(b)

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Write out the truth table for your symbolic statement.
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(c)

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You visit Joe’s residence room and found that Joe is unhappy even though the game is on. Assuming that the above conditional statement is a true statement about Joe, what can you conclude about the popcorn? Which rows in your truth table justify this conclusion?
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Activity 1.2.

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Consider the logical statement
(p→q)→(¬p∨q).
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(a)

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Make up an English language statement that has the same logical structure as this symbolic statement. (Do not just make a word-for-word translation of the logical connectives — make sure you have a reasonable-sounding English sentence when read out loud.)
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(b)

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Argue convincingly that this symbolic statement is a tautology, not by writing out its truth table, but by arguing that it is not possible for the statement to be false.
Hint.
Start with the assumption that this conditional statement is false, and then work backwards from the statement to the possible truth values of p and q based on that assumption to conclude that the statement being false is not actually possible.
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Activity 1.3.

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Consider the logical statement
(p∧¬r)→[(p→q)→(p∧¬r)].
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(a)

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Make the statement simpler by assigning new variables to represent compound statements and rewriting the statement in terms of the new variables.
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(b)

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Argue that your new statement is a tautology. What does this mean about the original statement?
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Activity 1.4.

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First, re-familiarize yourself with what it means when one statement logically implies another.
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Suppose that A logically implies B and B logically implies C. Must A logically imply C? Argue convincingly in support of your answer by arguing that the technical definition of logically implies is satisfied.
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