# Constructing truth tables and interpreting logic statements

I am asked to help set up a study group using sample study questions that gives some of us in the group the most trouble and I need help formulating these types of equations.

1. Determine the truth value of the following statement:

The Leaning Tower of Pisa is located in England and all prime numbers divisible by 1.

True or False

2. Construct a truth table for (p V q) → ~p

3. Fill in the heading of the following truth table using any of p, q, ~, →, ↔, V, and Λ.

P Q XXXXXXXX

T T F

T F T

F T F

F F F

4. Construct a truth table for ~p → (~p V q)

5. Given p is true, q is true, and r is false, find the truth value of the statement ~q → (~p Λ r). Show step by step work.

6. Determine which, if any, of the three statements are equivalent.

I) If the pipe is leaking, then I will not call the roofer.

II) Either the pipe is leaking or I will call the roofer.

III) If the pipe is not leaking, then I will call the roofer.

I and II are equivalent

II and III are equivalent

I and III are equivalent

I, II, and III are equivalent

None are equivalent

7. Write the argument below in symbols to determine whether it is valid or invalid. State a reason for your conclusion. Specify the p and q you used.

Either the gazebo is made of wood or the vine is growing on the gazebo.

The gazebo is not made of wood.

∴ The vine is growing on the gazebo.

P: The gazebo is made of wood.

Q: The vine is growing on the gazebo.

#### Solution Preview

1. Determine the truth value of the following statement:

The Leaning Tower of Pisa is located in England and all prime numbers divisible by 1.

True or False

The answer is false.

Problems like these can be created by making up two statements that you can declare as true or false. Then you can create any kind of compound statement by joining the two with "and", "or", "if ... then", and negations of either or both statement. You can determine the truth value of any such statement by making a truth table.

Example

P: The earth is flat.

Q: Beethoven was deaf.

What is the truth value of P → ~Q

P Q ~Q P → ~Q

T T F F

T F T T

F T F T

F F T T

2. Construct a truth table for (p V q) → ~p

P Q P V Q ~P (p V q)→~p

T T T F F

T F T F F

F T T T T

F F F T T

Truth tables are created by finding all four combinations of T and F for two statements. Then you can combine those in compound statements such as P V Q or ~P Λ Q. Any combination of Ps and Qs will work along with any implications. ...

#### Solution Summary

Constructing truth tables and interpreting logic statements.