QuestionCreate a truth table to show that $\quad[(p \vee q) \rightarrow r] \equiv(p \rightarrow r) \wedge(q \rightarrow r)$ .
Explain how the truth table shows these are logically equivalent.

Studdy Solution

STEP 1

What is this asking? We need to build a truth table to prove two logical statements mean the same thing: "If either p or q is true, then r is true" is the same as "If p is true, then r is true, AND if q is true, then r is true". Watch out! Don't rush the truth table!
Each row represents a different possibility, so make sure to carefully consider each one.
Remember, p → q is only false when p is true and q is false!

STEP 2

1. Set up the truth table
2. Evaluate the left side
3. Evaluate the right side
4. Compare the results

STEP 3

Alright, let's kick things off by setting up our truth table's framework.
We've got three variables: p, q, and r.
Each can be either true (T) or false (F), so we'll have $2 \cdot 2 \cdot 2 = 2^3 = 8$ different combinations to consider.
Let's list all those possibilities for p, q, and r.

STEP 4

Now, we'll add columns for $(p \vee q)$ , $(p \vee q) \rightarrow r$ , $(p \rightarrow r)$ , $(q \rightarrow r)$ , and finally, $(p \rightarrow r) \wedge (q \rightarrow r)$ .
This gives us a nice organized table to work with!
$\begin{array}{ccc} p & q & r & (p \vee q) & (p \vee q) \rightarrow r & (p \rightarrow r) & (q \rightarrow r) & (p \rightarrow r) \wedge (q \rightarrow r) \\ T & T & T & & & & & \\ T & T & F & & & & & \\ T & F & T & & & & & \\ T & F & F & & & & & \\ F & T & T & & & & & \\ F & T & F & & & & & \\ F & F & T & & & & & \\ F & F & F & & & & & \\ \end{array}$

STEP 5

Let's tackle the left side of our equivalence: $(p \vee q) \rightarrow r$ .
First, we need to figure out when $(p \vee q)$ is true.
Remember, "or" means at least one of p or q (or both!) must be true.
Let's fill in that column!

STEP 6

Now, we can evaluate $(p \vee q) \rightarrow r$ .
Remember, an implication is only false when the hypothesis (the left part) is true, and the conclusion (the right part) is false.

STEP 7

Time to conquer the right side: $(p \rightarrow r) \wedge (q \rightarrow r)$ .
We'll start by evaluating $(p \rightarrow r)$ .
Remember the rule: this is only false when p is true and r is false.

STEP 8

Next up, we'll figure out $(q \rightarrow r)$ .
Same rule applies: it's only false when q is true and r is false.

STEP 9

Finally, we'll combine our results to find $(p \rightarrow r) \wedge (q \rightarrow r)$ . "And" means both parts must be true for the whole thing to be true.

STEP 10

Now, the moment of truth!
Let's compare the columns for $(p \vee q) \rightarrow r$ and $(p \rightarrow r) \wedge (q \rightarrow r)$ .
If they match exactly, then we've proven our logical equivalence!
$\begin{array}{ccc|c|c|c|c|c} p & q & r & (p \vee q) & (p \vee q) \rightarrow r & (p \rightarrow r) & (q \rightarrow r) & (p \rightarrow r) \wedge (q \rightarrow r) \\ T & T & T & T & T & T & T & T \\ T & T & F & T & F & F & F & F \\ T & F & T & T & T & T & T & T \\ T & F & F & T & F & F & T & F \\ F & T & T & T & T & T & T & T \\ F & T & F & T & F & T & F & F \\ F & F & T & F & T & T & T & T \\ F & F & F & F & T & T & T & T \\ \end{array}$

STEP 11

The truth table shows that $[(p \vee q) \rightarrow r]$ and $(p \rightarrow r) \wedge (q \rightarrow r)$ have the same truth values for all possible combinations of p, q, and r.
Therefore, they are logically equivalent!

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