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 \(a = c\text{.}\) 

1. Write out the converse, inverse, and contrapositive of the above statement.

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

3. 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 \(B = (p\rightarrow q) \land (q \rightarrow p)\text{.}\)

1. Argue that if \(A\) is false, then so is \(B\text{.}\)

Do not use the proposed equivalence above as part of your argument.

2. Argue that if \(B\) is false, then so is \(A\text{.}\)

Do not use the proposed equivalence above as part of your argument.

3. 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.