2.4: Activities
Activity \(\PageIndex{1}\)
Write an English language statement that has the logical form \(\neg (A \lor B)\text{.}\) Then write one that has the form \(\neg A \land \neg B\text{,}\) 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?
Activity \(\PageIndex{2}\)
What do you think DeMorgan's Laws would say about \(\neg (A \land B \land C)\text{?}\) Use propositional calculus to justify your answer.
Activity \(\PageIndex{3}\)
Recall that a pair of coordinates \((x,y)\) defines a point in the Cartesian plane.
Consider the following conditional statement.
If Cartesian points \((a,b)\) and \((c,d)\) are actually the same point, then \(a = c\text{.}\)
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Write out the converse, inverse, and contrapositive of the above statement.
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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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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.
Activity \(\PageIndex{4}\)
In this activity, we will justify the equivalence
\begin{equation*} p \leftrightarrow q \Leftrightarrow(p\rightarrow q) \land (q \rightarrow p) \text{.} \end{equation*}
So consider the statements \(A = p \leftrightarrow q\) and \(B = (p\rightarrow q) \land (q \rightarrow p)\text{.}\)
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Argue that if \(A\) is false, then so is \(B\text{.}\)
Do not use the proposed equivalence above as part of your argument.
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Argue that if \(B\) is false, then so is \(A\text{.}\)
Do not use the proposed equivalence above as part of your argument.
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Explain why the the two arguments in Task a and Task b, taken together, justify the equivalence \(A \Leftrightarrow B\text{.}\) Do this without making any further arguments about the truth values of \(p\) and \(q\text{.}\)
Activity \(\PageIndex{5}\)
Consider the statements \(p \rightarrow (q_1 \lor q_2)\) and \((p \land \neg q_1) \rightarrow q_2\text{.}\)
Use propositional calculus and substitution to show that these two statements are equivalent.