1.5: Determining whether an assertion is true

To put them all in one place, the truth tables for the connectives of Propositional Logic are repeated here: \[\begin{array}{c||c}
\mathcal{A} & \neg \mathcal{A} \\
\hline \mathrm{T} & \mathrm{F} \\
\mathrm{F} & \mathrm{T}
\end{array}\]
\[\begin{array}{c|c||c|c|c|c}
\mathcal{A} & \mathcal{B} & \mathcal{A} \& \mathcal{B} & \mathcal{A} \vee \mathcal{B} & \mathcal{A} \Rightarrow \mathcal{B} & \mathcal{A} \Leftrightarrow \mathcal{B} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\
\mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T}
\end{array}\]

Truth tables for the connectives of Propositional Logic.

Every advanced math student needs to be able to quickly reproduce all of these truth tables, without looking them up.

Using these tables, you should be able to decide whether any given assertion is true or false, for any particular values of its assertion letters. (In this , we often refer to assertion letters as “variables.”)

Example \(1.5.1\).

Assume \(A\) is true, \(B\) is false, and \(C\) is false. Is \((A \lor B) \Rightarrow (B \& \lnot C)\) true?

Solution

We have \[\begin{aligned} (A \lor B) \Rightarrow (B \& \lnot C) &\quad= \quad(\mathsf{T} \lor \mathsf{F}) \Rightarrow (\mathsf{F} \& \lnot \mathsf{F}) \\&\quad= \quad\mathsf{T} \Rightarrow (\mathsf{F} \& \mathsf{T}) \\&\quad= \quad\mathsf{T} \Rightarrow \mathsf{F} \\&\quad= \quad\mathsf{F} . \end{aligned}\] The assertion is not true.

What does this mean in English? Suppose, for example, that we have the symbolization key

A: Bill baked an apple pie,

B: Bill baked a banana pie,

C: Bill baked a cherry pie.

Also suppose Ellen tells us (maybe because she knows what ingredients Bill has):

If Bill baked either an apple pie or a banana pie, then he baked a banana pie, but did not bake a cherry pie.

Now, it turns out that \[\text{Bill baked an apple pie, but did not bake a banana pie, and did not bake a cherry pie.}\] Then the above calculation shows that Ellen was wrong; her assertion is false.

Example \(1.5.2\).

Assume \(A\) is true, \(B\) is false, and \(C\) is true. Is \((A \lor C) \Rightarrow \lnot (A \Rightarrow B)\) true?

Solution

We have \[\begin{aligned} (A \lor C) \Rightarrow \lnot (A \Rightarrow B) &\quad= \quad (\mathsf{T} \lor \mathsf{T}) \Rightarrow \lnot (\mathsf{T} \Rightarrow \mathsf{F}) \\& \quad= \quad \mathsf{T} \Rightarrow \lnot \mathsf{F} \\& \quad= \quad \mathsf{T} \Rightarrow \mathsf{T} \\& \quad= \quad \mathsf{T} . \end{aligned}\] The assertion is true.

Exercise \(1.5.3\).

Determine whether each assertion is true for the given values of the variables.

1. \((A \lor C) \Rightarrow \lnot (A \Rightarrow B)\)
   1. \(A\) is true, \(B\) is false, and \(C\) is false.
   2. \(A\) is false, \(B\) is true, and \(C\) is false.
2. \(\bigl(P \lor \lnot (Q \Rightarrow R) \bigr) \Rightarrow \bigl( (P \lor Q) \& R \bigr)\)
   1. \(P\), \(Q\), and \(R\) are all true.
   2. \(P\) is true, \(Q\) is false, and \(R\) is true.
   3. \(P\) is false, \(Q\) is true, and \(R\) is false.
   4. \(P\), \(Q\), and \(R\) are all false.
3. \(\bigl( (U \& \lnot V) \lor (V \& \lnot W) \lor (W \& \lnot U) \bigr)\) 4em \(\Rightarrow \lnot (U \& V \& W)\)
   1. \(U\), \(V\), and \(W\) are all true.
   2. \(U\) is true, \(V\) is true, and \(W\) is false.
   3. \(U\) is false, \(V\) is true, and \(W\) is false.
   4. \(U\), \(V\), and \(W\) are all false.
4. \((X \lor \lnot Y) \& (X \Rightarrow Y)\)
   1. \(X\) and \(Y\) are both true.
   2. \(X\) is true and \(Y\) is false.
   3. \(X\) is false and \(Y\) is true.
   4. \(X\) and \(Y\) are both false.