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Elementary Foundations: An introduction to topics in discrete mathematics

Jeremy Sylvestre

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Activities 2.4 Activities
A4US

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Activity 2.1.

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Write an English language statement that has the logical form ¬(A∨B). Then write one that has the form ¬A∧¬B, where A and B are the same as in your first sentence. DeMorgan’s Laws say your two sentences are logically equivalent. Do you agree?
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Activity 2.2.

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What do you think DeMorgan’s Laws would say about ¬(A∧B∧C)? Use propositional calculus to justify your answer.
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Activity 2.4.

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Recall that a pair of coordinates (x,y) defines a point in the Cartesian plane.
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Consider the following conditional statement.
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If Cartesian points (a,b) and (c,d) are actually the same point, then a=c.
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(b)

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You now have four conditional statements. For each of the four, decide whether it is true, and justify your answer.
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(c)

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For each of the three new conditional statements from Task a in turn, take the view that that statement is the original conditional, and decide which of the others are its converse, inverse, and contrapositive.
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Activity 2.5.

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In this activity, we will justify the equivalence
p↔q⇔(p→q)∧(q→p).
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So consider the statements A=p↔q and B=(p→q)∧(q→p).
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(a)

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Argue that if A is false, then so is B.
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Do not use the proposed equivalence above as part of your argument.
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(b)

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Argue that if B is false, then so is A.
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Do not use the proposed equivalence above as part of your argument.
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(c)

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Explain why the two arguments in Task a and Task b, taken together, justify the equivalence A⇔B. Do this without making any further arguments about the truth values of p and q.
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Activity 2.6.

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Consider the statements p→(q1∨q2) and (p∧¬q1)→q2.
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Use propositional calculus and substitution to show that these two statements are equivalent.
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