Truth Tables
Week 7 Worksheet Seven
Topic: Truth Tables
Today we will examine a technique that helps us determine the truth of premises and ultimately the validity of arguments.
This technique involves drawing a table. For example, like this:
| S | M | |
Let’s say the letter S in the first column stands for Amy majors in Science, and the letter M, in the second column, stands for Amy majors in Mathematics.
Now let’s also put in the third column of this table the expression S v M, which will stand for Amy majors in Science or Mathematics, like this:
| S | M | S Ú M |
Now let’s imagine you do not remember what Amy majors in, but your friend claims to remember what she majors in and says: “Amy majors in Science or she majors in Math.”
However, you want to verify whether your friend is telling you the truth, so you decide to call Amy but before you do that you realize that there are four distinct possibilities or scenarios of you will find out:
- She majors in both!
- She majors in Science and not in Math.
- She does not major in Science but she majors in Math.
- She does not major in either of these.
Let’s now enter this information into our table.
In the first row, we will entre T under S and T under M because in the first scenario it is True that she majors in both.
| S | M | S Ú M |
| T | T | |
Then we enter the remaining truth values according to all the scenarios, like this:
| S | M | S Ú M |
| T | T | |
| T | F | |
| F | T | |
| F | F |
What truth values should we enter in the third column?
In the first situation, where Amy double-majors, we can say that your friend was telling you the Truth. It would make no sense to accuse your friend of lying to you just because he said: “Amy majors in Science or in Math.” So we enter T for that claim in the first row. Like this:
| S | M | S Ú M |
| T | T | T |
| T | F | |
| F | T | |
| F | F |
Your friend is only lying in a situation when Amy does not major in these two subjects at all. So we entered F the 4th row.
Your friend is also telling you the Truth in the second scenario, when it is true that she majors in Science but does not major in Math. Thus, we entered T in the second row.
And likewise, your friend is telling you the truth in the case, when Amy does not major in Science but she majors in Math. So we enter the truth values like this:
| S | M | S Ú M |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Now your turn. Create a disjunction, meaning a sentence that has the word “or.”
But make the choices not mutually exclusive. (*Mutually exclusive means that you can’t have it both ways as in My dog is dead and alive at the same time, or today is Monday and Tuesday at the same time.) So make the two choices possible to exist at the same time. In the previous example, it was possible to major in both fields. Again, think of two things that you can have at the same time, such as walking and talking or eating and listening to music.
Then, list all 4 possibilities/scenarios for that sentence. (Just like in the example above.)
Your sentence in English:
Maggie is eating an apple or a banana
Your sentence is symbols:
A v B
4 possible scenarios for that sentence:
- Maggie is eating both fruits
- Maggie is eating a banana and not an apple
- Maggie is not eating a banana but is eating an apple
- Maggie is not eating either of the fruits
Now enter the truth values into the table. Enter also the letters.
| A | B | A v B |
| T
T F F |
T
F T F |
T
T T F |
If you had difficulties with this the above assignment, go over this example:
Let’s say the letter A stands for: “John majors in Accounting.”
Let’s also say the letter B stands for” John majors in Biology.”
The sentence: “John majors in Accounting or in Biology” will have the following truth table:
| A | B | A Ú B |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
But let’s suppose that the sentence is not a disjunction but a conjunction, like this:
“John majors in Accounting and in Biology.”
Will the truth table be exactly the same? No.
When will this sentence be true? In what scenarios is this conjunction True? This sentence can only be true when John really majors in both of these subjects. In any other scenario this sentence would be a lie.
Thus, the truth table for conjunction looks like this:
| A | B | A & B |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
What about this sentence:
“There is life on other planets in the Milky Way and there are also aliens in the Andromeda galaxy.”
Let’s symbolize this conjunction as M & A
When is this sentence true?
Let’s list all the possible scenarios:
- There is life on other planets in the Milky Way and there are also aliens in the Andromeda galaxy.
- There is life on other planets in the Milky Way but there is no life in the Andromeda galaxy.
- There is no life on other planets in our galaxy but there is life in the Andromeda galaxy.
- There is no other life in the Milky Way and also no life in the Andromeda galaxy.
The truth values for this conjunction will be exactly the same as the truth values for the conjunction above:
| M | A | M&A |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Now your turn. Create a conjunction, meaning a sentence that has the word “and.”
Then, list all 4 possibilities/scenarios for that sentence.
Your sentence in English:
Anne studies arts and literature
Your sentence is symbols:
A & L
4 possible scenarios for that sentence:
- Anne studies both
- Anne studies arts but not literature
- Anne does not study arts but studies literature
- Anne does not study either subjects
Now enter the truth values reflecting the 4 scenarios you just listed:
| A | L | A & B |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
What about this sentence:
“John has a Girlfriend or he has a Wife and Children.”
In symbols: G v (W & C)
*The truth table for this sentence requires that we list more than 4 scenarios!
It requires that we consider 8 different scenarios, like this:
| G | W | C | W&C | G v (W&C) |
| T | T | T | ||
| T | T | F | ||
| T | F | T | ||
| T | F | F | ||
| F | T | T | ||
| F | T | F | ||
| F | F | T | ||
| F | F | F |
If this is puzzling, consider making a list all the possible results when you have to toss a coin 3 times in a row. Lets’ make a list of all the possible outcomes of Heads (H) and Tails (T).
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
How many outcomes are there?
EIGHT
In our sentence “John has a Girlfriend or he has a Wife and Children” we have 3 variables (girlfriend wife, children), we will have to create a truth table for 8 possible outcomes:
To make this easier first fill out the truth values for the conjunction: Wife and Children (W & C):
Then you can move on to the column: G v (W&C):f
| G | W | C | W&C | G v (W&C) |
| T | T | T | T | T v T = T |
| T | T | F | F | T v F = T |
| T | F | T | F | T v F = T |
| T | F | F | F | T v F = T |
| F | T | T | T | F v T = T |
| F | T | F | F | F v F = F |
| F | F | T | F | F v F = F |
| F | F | F | F | F v F = F |
Please enter all the truth values into the table above!
Lastly, let’s consider a truth table for negation.
Let’s say the letter M stands for the sentence “Today is Monday”.
If Today is indeed Monday, then the sentence is true, we will enter T in the first column, first row.
If today is not Monday we will enter F in the second row, first column, like this:
| M | |
| T | |
| F |
But now what will be the truth values for – M? Let’s enter – M in the second column, like this:
| M | – M |
| T | |
| F |
Negative M will be false, when positive M is true. Since -M stands for “It’s not Monday” but you are saying this on a Monday. So you are lying. Thus we enter F, like this:
| M | – M |
| T | F |
| F |
But if you say “It is not Monday” and you say that sentence on Friday for example, then you are saying something True, thus we enter T:
| M | – M |
| T | F |
| F | T |
Now your turn.
Enter the truth values for the sentence: “It’s raining.”
| R | – R |
| T | F |
| F | T |
Enter the truth values for the sentence: “Justice exists in this world.”
| E | – E |
| T | F |
| F | T |
Enter the truth values for the conjunction: S & – W
Let’s say S & – W stands for “Today is Sunday (S) and I do not have to work (- W).”
First let’s list the possible truth values for S and the possible truth values for E.
| S | W | S & – W |
| T | T | |
| T | F | |
| F | T | |
| F | F |
Now we will enter truth values in the third column, the one that says S & – E .
| S | W | S & – W |
| T | T | T & F=F |
| T | F | T& T=T |
| F | T | F&F=F |
| F | F | F&T=F |
Now your turn; complete these truth tables.
(The first row is already given in red)
| X | Y | X v Y |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
| X | Y | – X v Y |
| T | T | F v T= T |
| T | F | F v F = F |
| F | T | T v T = T |
| F | F | T v F = T |
| X | Y | X & Y |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
| X | Y | – X & Y |
| T | T | F&T=F |
| T | F | F & F = F |
| F | T | T & T =T |
| F | F | T & F = F |
| A | B | – ( A & B) & ( A & B) |
| T | T | -(T & T) & (T & T)
-T & T = F & T = F |
| T | F | -(T & F) & (T & F)
-F & F = F & F = F |
| F | T | -(F & T) & (F & T)
– F & F = F & F = F |
| F | F | -(F & F) & (F & F)
– F & F = F & F = F |
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Question 
Week 7 Worksheet Seven
Topic: Truth Tables
Today we will examine a technique that helps us determine the truth of premises and ultimately the validity of arguments.
Truth Tables
This technique involves drawing a table. For example, like this:
| S | M | |
Let’s say the letter S in the first column stands for Amy majors in Science, and the letter M, in the second column, stands for Amy majors in Mathematics.
Now let’s also put in the third column of this table the expression S v M, which will stand for Amy majors in Science or Mathematics, like this:
| S | M | S Ú M |
Now let’s imagine you do not remember what Amy majors in, but your friend claims to remember what she majors in and says: “Amy majors in Science or she majors in Math.”
However, you want to verify whether your friend is telling you the truth, so you decide to call Amy but before you do that you realize that there are four distinct possibilities or scenarios of you will find out:
- She majors in both!
- She majors in Science and not in Math.
- She does not major in Science but she majors in Math.
- She does not major in either of these.
Let’s now enter this information into our table.
In the first row, we will entre T under S and T under M because in the first scenario it is True that she majors in both.
| S | M | S Ú M |
| T | T | |
Then we enter the remaining truth values according to all the scenarios, like this:
| S | M | S Ú M |
| T | T | |
| T | F | |
| F | T | |
| F | F |
What truth values should we enter in the third column?
In the first situation, where Amy double-majors, we can say that your friend was telling you the Truth. It would make no sense to accuse your friend of lying to you just because he said: “Amy majors in Science or in Math.” So we enter T for that claim in the first row. Like this:
| S | M | S Ú M |
| T | T | T |
| T | F | |
| F | T | |
| F | F |
Your friend is only lying in a situation when Amy does not major in these two subjects at all. So we entered F the 4th row.
Your friend is also telling you the Truth in the second scenario, when it is true that she majors in Science but does not major in Math. Thus, we entered T in the second row.
And likewise, your friend is telling you the truth in the case, when Amy does not major in Science but she majors in Math. So we enter the truth values like this:
| S | M | S Ú M |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Now your turn. Create a disjunction, meaning a sentence that has the word “or.”
But make the choices not mutually exclusive. (*Mutually exclusive means that you can’t have it both ways as in My dog is dead and alive at the same time, or today is Monday and Tuesday at the same time.) So make the two choices possible to exist at the same time. In the previous example, it was possible to major in both fields. Again, think of two things that you can have at the same time, such as walking and talking or eating and listening to music.
Then, list all 4 possibilities/scenarios for that sentence. (Just like in the example above.)
Your sentence in English:
Your sentence is symbols:
4 possible scenarios for that sentence:
1.
2.
3.
Now enter the truth values into the table. Enter also the letters.
If you had difficulties with this the above assignment, go over this example:
Let’s say the letter A stands for: “John majors in Accounting.”
Let’s also say the letter B stands for” John majors in Biology.”
The sentence: “John majors in Accounting or in Biology” will have the following truth table:
| A | B | A Ú B |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
But let’s suppose that the sentence is not a disjunction but a conjunction, like this:
“John majors in Accounting and in Biology.”
Will the truth table be exactly the same? No.
When will this sentence be true? In what scenarios is this conjunction True? This sentence can only be true when John really majors in both of these subjects. In any other scenario this sentence would be a lie.
Thus, the truth table for conjunction looks like this:
| A | B | A & B |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
What about this sentence:
“There is life on other planets in the Milky Way and there are also aliens in the Andromeda galaxy.”
Let’s symbolize this conjunction as M & A
When is this sentence true?
Let’s list all the possible scenarios:
- There is life on other planets in the Milky Way and there are also aliens in the Andromeda galaxy.
- There is life on other planets in the Milky Way but there is no life in the Andromeda galaxy.
- There is no life on other planets in our galaxy but there is life in the Andromeda galaxy.
- There is no other life in the Milky Way and also no life in the Andromeda galaxy.
The truth values for this conjunction will be exactly the same as the truth values for the conjunction above:
| M | A | M&A |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Now your turn. Create a conjunction, meaning a sentence that has the word “and.”
Then, list all 4 possibilities/scenarios for that sentence.
Your sentence in English:
Your sentence is symbols:
4 possible scenarios for that sentence:
1.
2.
3.
Now enter the truth values reflecting the 4 scenarios you just listed:
What about this sentence:
“John has a Girlfriend or he has a Wife and Children.”
In symbols: G v (W & C)
*The truth table for this sentence requires that we list more than 4 scenarios!
It requires that we consider 8 different scenarios, like this:
| G | W | C | W&C | G v (W&C) |
| T | T | T | ||
| T | T | F | ||
| T | F | T | ||
| T | F | F | ||
| F | T | T | ||
| F | T | F | ||
| F | F | T | ||
| F | F | F |
If this is puzzling, consider making a list all the possible results when you have to toss a coin 3 times in a row. Lets’ make a list of all the possible outcomes of Heads (H) and Tails (T).
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
How many outcomes are there?
In our sentence “John has a Girlfriend or he has a Wife and Children” we have 3 variables (girlfriend wife, children), we will have to create a truth table for 8 possible outcomes:
To make this easier first fill out the truth values for the conjunction: Wife and Children (W & C):
Then you can move on to the column: G v (W&C):
| G | W | C | W&C | G v (W&C) |
| T | T | T | T | T v T = T |
| T | T | F | ||
| T | F | T | ||
| T | F | F | ||
| F | T | T | ||
| F | T | F | ||
| F | F | T | ||
| F | F | F |
Please enter all the truth values into the table above!
Lastly, let’s consider a truth table for negation.
Let’s say the letter M stands for the sentence “Today is Monday”.
If Today is indeed Monday, then the sentence is true, we will enter T in the first column, first row.
If today is not Monday we will enter F in the second row, first column, like this:
| M | |
| T | |
| F |
But now what will be the truth values for – M? Let’s enter – M in the second column, like this:
| M | – M |
| T | |
| F |
Negative M will be false, when positive M is true. Since -M stands for “It’s not Monday” but you are saying this on a Monday. So you are lying. Thus we enter F, like this:
| M | – M |
| T | F |
| F |
But if you say “It is not Monday” and you say that sentence on Friday for example, then you are saying something True, thus we enter T:
| M | – M |
| T | F |
| F | T |
Now your turn.
Enter the truth values for the sentence: “It’s raining.
| R | – R |
Enter the truth values for the sentence: “Justice exists in this world.”
| E | – E |
Enter the truth values for the conjunction: S & – W
Let’s say S & – W stands for “Today is Sunday (S) and I do not have to work (- W).”
First let’s list the possible truth values for S and the possible truth values for E.
| S | W | S & – W |
| T | T | |
| T | F | |
| F | T | |
| F | F |
Now we will enter truth values in the third column, the one that says S & – E .
| S | W | S & – W |
| T | T | T & F=F |
| T | F | T& T=T |
| F | T | F&F=F |
| F | F | F&T=F |
Now your turn; complete these truth tables.
(The first row is already given in red)
| X | Y | X v Y |
| T | T | T |
| T | F | |
| F | T | |
| F | F |
| X | Y | – X v Y |
| T | T | F v T= T |
| T | F | |
| F | T | |
| F | F |
| X | Y | X & Y |
| T | T | T |
| T | F | |
| F | T | |
| F | F |
| X | Y | – X & Y |
| T | T | F&T=F |
| T | F | |
| F | T | |
| F | F |
| A | B | – ( A & B) & ( A & B) |
| T | T | |
| T | F | |
| F | T | |
| F | F |