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Readability settings Prev Up Next \(\require{cancel}
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Activities 2.4 Activities
Activity 2.1 .
Write an English language statement that has the logical form
\(\lgcnot (A \lgcor B) \text{.}\) Then write one that has the form
\(\lgcnot A \lgcand \lgcnot B \text{,}\) 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 2.2 .
What do you think
DeMorgan’s Laws would say about
\(\lgcnot (A \lgcand B \lgcand C) \text{?}\) Use propositional calculus to justify your answer.
Activity 2.3 .
Activity 2.4 .
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
\(a = c \text{.}\)
(a) (b) You now have four conditional statements. For each of the four, decide whether it is true, and justify your answer.
(c) 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 2.5 .
In this activity, we will justify the equivalence
\begin{equation*}
p \lgcbicond q \lgcequiv (p\lgccond q) \lgcand (q \lgccond p) \text{.}
\end{equation*}
So consider the statements \(A = p \lgcbicond q \) and \(B = (p\lgccond q) \lgcand (q \lgccond p) \text{.}\)
(a) Argue that if
\(A \) is false, then so is
\(B \text{.}\)
Do
not use the proposed equivalence above as part of your argument.
(b) Argue that if
\(B \) is false, then so is
\(A \text{.}\)
Do
not use the proposed equivalence above as part of your argument.
(c) Explain why the two arguments in
Task a and
Task b , taken together, justify the equivalence
\(A \lgcequiv B \text{.}\) Do this
without making any further arguments about the truth values of
\(p \) and
\(q \text{.}\)
Activity 2.6 .
Consider the statements
\(p \lgccond (q_1 \lgcor q_2) \) and
\((p \lgcand \lgcnot q_1) \lgccond q_2 \text{.}\)
Use propositional calculus and substitution to show that these two statements are equivalent.
