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# Logic

Logic
• Calculus • Decision Maths • Geometry • Mechanics Maths • Probability and Statistics • Pure Maths • Statistics Three logicians walk into a bar. The bartender says "would you all like a drink?" The first logician answers "I'm not sure." The second logician also answers "I'm not sure." The third and final logician answers "yes."

This joke requires some knowledge of mathematical logic to understand. The first and second logician's wanted a drink but did not know if all three of them would like a drink, which is what the bartender asked, and hence answered "I don't know". If either of these men did not want a drink, they would have instead answered "no", since in this scenario it would not be the case that all three men want a drink. The third man, knowing that if either of the other two men didn't want a drink they would have answered "no", can deduce that all three of them want a drink and thus answers "yes". And who said math couldn't be fun?

## Mathematical Logic Meaning

Logic is essentially the study of truth and reasoning. Logic is used in everyday life all the time.

For example, given the statements:

• If it is raining, I will stay at home.

• If I'm at home, I will wear my slippers.

From these statements alone, if you see it raining you are able to conclude that I will be wearing my slippers, even though I never directly told you this. This is an example of logical reasoning.

Mathematical Logic is the application of logic to Mathematics.

## Types of Logic

There are many different types of logic, but the two most fundamental branches of mathematical logic are:

• Propositional Logic (Also called Propositional Calculus),

• First-Order Logic.

First-order logic is an extension of propositional logic, with a few extra components that allow you to apply the ideas of propositional logic to many other areas of mathematics. For high school mathematics, only propositional logic is needed.

### Propositional Logic

Propositional logic, as the name would suggest, is the study of mathematical propositions.

Let's define what a proposition is.

A proposition is a statement that is either true or false.

Some examples of mathematical propositions are:

• 3 is odd.
• 3 is even.
• 7 + 4 = 12.
• 1 is a prime number.

These statements all must be true or false, they cannot be both and they cannot be neither. In propositional logic, there is no 'maybe' or 'sort of'.

#### Connectives

Propositions can be joined using connectives.

Connectives join propositional statements together, into a larger propositional statement.

Connectives are used all the time in both everyday life and mathematics. For example:

• 3 is a prime number and 3 is odd. The connective here is "and".

• 3 is a prime number or 2 is odd. The connective here is "or".

• It is not true that 7 + 4 = 12. The connective here is "not".

• 6 is even implies that 6 divisible by 2. The connective here is "implies". This can also be thought of as an "If... then..." statement.

Connectives can be combined to make more complex statements. Consider Pythagoras' Theorem:

"If $$a, b, c$$ are the sides of a right-angle triangle and $$c$$ is the hypotenuse, then $$a^2 + b^2 = c^2.$$"

This uses the connective "and" as well as the connective "if... then..." or "implies".

The main connectives in propositional logic are:

 Connective Symbol And $$\land$$ Or $$\lor$$ Not $$\lnot$$ Implies $$\implies$$

Many different, and more complicated, propositions can be made using these connectives.

Rather that writing out sentences, mathematicians often use a letter in place of a proposition. If $$p, q, r, s$$ are propositions, then the following are also propositions:

• $$(p \land q):$$ $$p$$ and $$q.$$
• $$(p \lor q):$$ $$p$$ or $$q.$$
• $$(\lnot p):$$ not $$p.$$
• $$(p \implies q):$$ $$p$$ implies $$q.$$
• $$((p \land q) \implies r):$$ if $$p$$ and $$q,$$ then $$r.$$
• $$((p \land (\lnot q)) \implies (r \land s))$$ if $$p$$ and not $$q,$$ then $$r$$ and $$s.$$

#### Truth Tables

To understand the meaning of these connectives, truth tables are used.

A truth table is a table showing when a more complicated proposition is true, based on when each individual component of it is true.

Let's first look at the truth table for 'and'. The statement "$$p$$ and $$q$$" is only true if $$p$$ is true and $$q$$ is true, hence this is the only case in which the column for $$p \land q$$ is true.

 $$p$$ $$q$$ $$p \land q$$ True True True True False False False True False False False False

The truth table for 'or' is below. The statement "$$p$$ or $$q$$" is only true if $$p$$ is true or $$q$$ is true. In other words, if one of the statements is true, then the or statement is true.

 $$p$$ $$q$$ $$p \lor q$$ True True True True False True False True True False False False

'Not' essentially changes the value from true to false, or false to true. The truth table for 'not' is:

 $$p$$ $$\lnot p$$ True False False True

The truth table for 'implies' is:

 $$p$$ $$q$$ $$p \implies q$$ True True True True False False False True True False False True

The truth tables for or and not are quite intuitive, but the truth table for implies can look quite strange at first. To understand it, think of the following statement: "If you can do a backflip, I'll pay you \$20." The possibilities here are:

• You manage to do the backflip, and I give you the money. In this case, the statement was true.

• You manage to do the backflip, and I don't give you the money. In this case, the statement was false.

• If you fail to do the backflip, it doesn't matter if I pay you or not, because I have still kept to my word so far. Hence, the statement is still true.

Truth tables can be made for more complicated statements. You can do this by increasing the number of columns, having columns dedicated to each section of the statement with increasing size each time. The truth table for one of the statements you saw earlier, $$((p \land q) \implies r)$$, looks like this:

 $$p$$ $$q$$ $$r$$ $$p \land q$$ $$((p \land q) \implies r)$$ True True True True True True True False True False True False True False True True False False False True False True True False True False True False False True False False True False True False False False False True

As you can see, truth tables grow very quickly. With each new letter added, the number of rows will double. This is why a different format is often used, called Logic Trees.

#### Logic Trees

A logic tree is another way of showing when a proposition is true.

A logic tree is a tree diagram for showing when a proposition is true. Each branching point is when one of the variables is determined to be true or false.

Let's first look at the logic tree for '$$p$$ and $$q$$', written $$p \land q$$. The logic tree for $$p \land q$$.

You will see that at the top of this tree, there is the letter $$p.$$ This is called the root node, meaning that it is the first point in the tree, where everything else branches off from, just like the root of a tree. This represents the first decision that has to be made for this proposition: is $$p$$ true or false?

Underneath this, there are arrows pointing downwards. These are called branches. Each one of these represents an answer to the decision from the previous node above it. The branch on the left represents $$p$$ being true, and the branch on the right represents $$p$$ being false. This is shown by the labels on the sides of the arrows.

In the situation where $$p$$ is false, you know for a fact that $$p \land q$$ will also be false, no matter what $$q$$ is. This is why underneath the 'false' branch, there is an 'F'. This is called a leaf node, and represents the value of the whole proposition.

If you follow the 'true' branch instead, you will reach the letter $$q.$$ This is called a decision node. This is just like the root node, but instead represents the decision: is $$q$$ true or false? Since $$p$$ being true is not enough to determine for a fact whether or not $$p \land q$$ will be true, you must also look into the case where $$q$$ is true and $$q$$ is false, hence the branches underneath this decision node. Since at this point both of the variables ($$p$$ and $$q$$) in the proposition have been explored, the nodes underneath $$q$$ must be leaf nodes, as there are no more decisions to be made.

So far, you know how to create the logic tree. But how do you read the tree? Starting at the top!

• If $$p$$ is false, go down the 'F' branch. You reach an 'F' leaf node, so the proposition must be false if $$p$$ is false.

• If $$p$$ is true, you go down the $$T$$ branch to reach the decision node for $$q.$$ This makes sense, as it is still unclear whether or not $$p \land q$$ is true or false yet.

• If from this point, you choose $$q$$ to be true, then you go down the $$T$$ branch again and will reach a $$T$$ leaf node. This means in this case, the proposition must be true.

• If from the $$q$$ decision node, you instead choose false, you will go down the $$F$$ branch instead and reach an $$F$$ leaf node. In this case, the proposition must also be false.

Before you look at the logic trees for the other connectives, lets look at a quick example where you must build a logic tree and a truth table side by side using the slippers and rain example from the beginning, to see that truth tables and logic trees represent the same thing.

Make a truth table and logic tree for the following statement:

"If it is raining, I will wear my slippers."

Solution

There are two individual propositions in this statement:

• It is raining.
• I am wearing my slippers.

These will be the first two columns in the truth table, and the two decision nodes in the logic tree. Let's begin with the truth table. The truth table will look like this with the first two columns filled in:

 Raining. Slippers. If it is raining, I will wear my slippers. True True True False False True False False

If it is raining and I am wearing my slippers, the main proposition must be true. Hence, the first row in the "If it is raining, I will wear my slippers" column must be true.

 Raining. Slippers. If it is raining, I will wear my slippers. True True True True False False True False False

If it is raining and I am not wearing my slippers, the proposition must be false. Hence, the second row in the main proposition column must be false.

 Raining. Slippers. If it is raining, I will wear my slippers. True True True True False False False True False False

As discussed earlier when the truth table for "implies" was shown, no matter what slippers is, if raining is false then the proposition will be false. For this reason, the last two columns will be false.

 Raining. Slippers. If it is raining, I will wear my slippers. True True True True False False False True True False False True

You may notice that this is exactly the same as the truth table for $$p \implies q.$$ This makes sense, as this is exactly the same as the proposition $$p \implies q,$$ but where $$p$$ and $$q$$ have been given real-world meanings.

Now let's build the logic tree for this. The root node will be "It is raining." If it is raining but you do not know whether I am wearing slippers, you can see in the logic table that the proposition could be true or false. This means that the true branch coming off the root node must lead to a decision node for "I am wearing slippers." If it is not raining, you can see in the logic table that the proposition is always true. For this reason, the false branch of the root node must lead to a leaf node for true. The root node and first two branches of the proposition "If it is raining, I will wear my slippers."

Now, all that is left is to put in the branches and leaf nodes coming off the "I am wearing slippers." decision node. Since in this case "it is raining." is true, if "I am wearing slippers." is true, the whole proposition must be true. Hence, the true branch must lead to a true leaf node. If "I am wearing slippers" is false, the whole proposition must be false, meaning the false branch must lead to a false leaf node. The final logic tree will look like this: The logic tree for the proposition "If it raining, I will wear my slippers."

You should notice how no matter which assignment of "truth" and "false" you put for each proposition, the outcome is the same in the logic tree and truth table.

Now that you have seen an example of a logic tree side by side with a truth table, let's look at the logic trees for the other connectives.

The logic tree for '$$p$$ or $$q$$', written '$$p \lor q$$', is: The logic tree for $$p \lor q$$.

The logic tree for 'not $$p$$', written $$\lnot p$$, is: The logic tree for $$\lnot p.$$

The logic tree for '$$p$$ implies $$q$$', written $$p \implies q$$, is: The logic tree for $$p \implies q$$.

These logic trees all come quite naturally from the truth tables.

Using the same example from the previous section with 3 initial propositions, $$((p \land q) \implies r),$$ you will see that logic trees can greatly simplify the amount of writing required. This is the logic tree for $$((p \land q) \implies r):$$ The logic tree for $$((p \land q) \implies r)$$.

## Mathematical Logic Examples

First, let's look at some examples where you must work out the truth tables and logic trees of various propositions.

Finish the following truth table, for $$(p \land (\lnot q)).$$

 $$p$$ $$q$$ $$\lnot q$$ $$p \land \lnot q$$ True True True False False True False False

Solution

First, fill in the column for $$\lnot q.$$ Remember that whenever $$q$$ is true, $$\lnot q$$ will be false, and whenever $$q$$ is false, $$\lnot q$$ will be true.

 $$p$$ $$q$$ $$\lnot q$$ $$p \land \lnot q$$ True True False True False True False True False False False True

Now you can fill in $$p \land \lnot q.$$ Remember that this will only be true if $$p$$ is true and $$\lnot q$$ is true, and false everywhere else.

 $$p$$ $$q$$ $$\lnot q$$ $$p \land \lnot q$$ True True False False True False True True False True False False False False True False

Let's look at a similar example, but using a logic tree instead.

Draw a logic tree for the proposition $$(\lnot p) \implies q.$$

Solution

The first step, as always for logic trees, is the root node. The root node, in this case, will be $$p.$$ You can write a $$p$$ at the top of the diagram to represent it. This root node will have 2 branches underneath it, one representing the scenario where $$p$$ is true, and the other representing the scenario where $$p$$ is false. If $$p$$ is true, then $$\lnot p$$ must be false. $$\lnot p$$ is the left-hand side of the implication in our proposition, and as stated when you first looked at the 'implies' connective, if the left hand side of the implication is false, the whole proposition is false. For this reason, $$p$$ being false must lead to a 'false' leaf node.

If $$p$$ is false, $$\lnot p$$ must be true. Since $$\lnot p$$ is the left hand side of the implication, and this is true, it is impossibile to tell whether the implication as a whole is true or false, until $$q$$ has been decided. For this reason, $$p$$ being true must lead to a decision node for $$q.$$ The diagram so far will look like this: If $$p$$ is true, $$\lnot p \implies q$$ must be true. If $$p$$ is not true, it is yet unclear whether $$lnot p \implies q$$ is true.

If $$p$$ is true, $$\lnot p$$ will be false and hence the implication will always be true.

The remainder of this proposition is just like a normal implication. Hence, if $$q$$ is true, the implication will be true, meaning the 'true' branch must lead to a 'true' leaf node. If $$q$$ is false, the implication will also be false, meaning the 'false' branch must lead to a 'false' leaf node. The logic tree for $$(\lnot p) \implies q$$ will thus look like: The decision tree for $$\lnot p \implies q$$

The rest of the implication is the same as the result of a normal implication. Hence $$q$$ is true will make the result true, and $$q$$ is false will make the result false.

## Mathematical Logic Problems

Here, you will see some important properties within logic, and learn how to solve them using truth tables. Before this though, an important relation must be introduced.

Two propositions are equivalent if they are always the same, no matter what the individual variables are set as. If $$p$$ and $$q$$ are equivalent, then you write $$p \equiv q.$$

Don't think that equivalence is the same as 2 things being equal, they are very slightly different. For example, it is okay to say $$2x = x + 8,$$ as you interpret this to be true for a particular value of $$x$$ (in this case, $$x = 8$$.) However, you can't write $$2x \equiv x + 8,$$ because this is not true: $$2x$$ and $$x+8$$ are not the same thing, they are completely different functions.

Now, onto the properties of logical connectives. These results are:

• Double Negation: $\lnot (\lnot p) \equiv p.$

• Commutativity: \begin{align} p \land q & \equiv q \land p, \\ p \lor q & \equiv q \lor p. \end{align}
• Associativity: \begin{align} ( p \land (q \land r)) & \equiv ( (p \land q) \land r), \\ ( p \lor (q \lor r)) & \equiv ( ( p \lor q) \lor r). \end{align}

• Distributivity: \begin{align} ( p \land (q \lor r)) & \equiv ((p \land q ) \lor ( p \land r) ), \\ (p \lor (q \land r)) & \equiv ((p \lor q) \land (p \lor r)). \end{align}

These formulas can be proven by showing that their truth tables or their logic trees are the same. Let's look at an example of this.

Prove the associativity of $$\land:$$ $( p \land (q \land r)) \equiv ( (p \land q) \land r)$ by filling in the following truth table.

 $$p$$ $$q$$ $$r$$ $$q \land r$$ $$p \land q$$ $$( p \land (q \land r))$$ $$((p \land q) \land r)$$ True True True True True False True False True True False False False True True False True False False False True False False False

Solution

First, fill in the column for $$q \land r.$$ This will only be true when $$q$$ is true and $$r$$ is true, and will be false everywhere else. This finished column should look like:

 $$p$$ $$q$$ $$r$$ $$q \land r$$ $$p \land q$$ $$( p \land (q \land r))$$ $$((p \land q) \land r)$$ True True True True True True False False True False True False True False False False False True True True False True False False False False True False False False False False

Now, you can fill in the column for $$p \land q.$$ This works exactly the same as the last column you filled in: it will be true whenever $$p$$ is true and $$q$$ is true, and false everywhere else.

 $$p$$ $$q$$ $$r$$ $$q \land r$$ $$p \land q$$ $$( p \land (q \land r))$$ $$((p \land q) \land r)$$ True True True True True True True False False True True False True False False True False False False False False True True True False False True False False False False False True False False False False False False False

Now, you can fill in the column for $$( p \land (q \land r)).$$ This will only be true when the column for $$p$$ is true and the column for $$q \land r$$ is true, and will be false everywhere else.

 $$p$$ $$q$$ $$r$$ $$q \land r$$ $$p \land q$$ $$( p \land (q \land r))$$ $$((p \land q) \land r)$$ True True True True True True True True False False True False True False True False False False True False False False False False False True True True False False False True False False False False False False True False False False False False False False False False
Finally, you can fill in the last column. This will only be true when both $$p \land q$$ is true and $$r$$ is true, and will be false everywhere else.
 $$p$$ $$q$$ $$r$$ $$q \land r$$ $$p \land q$$ $$( p \land (q \land r))$$ $$((p \land q) \land r)$$ True True True True True True True True True False False True False False True False True False False False False True False False False False False False False True True True False False False False True False False False False False False False True False False False False False False False False False False False

Since the final two columns are the same, it must be the case that these two propositions are equivalent.

Next, let's look at a similar example but using a logic tree instead.

The logic tree for $$((p \land q ) \lor ( p \land r))$$ is: This is the logic tree of $$((p \land q ) \lor ( p \land r)).$$

Prove the distributivity of $$\land$$ over $$\lor:$$ $$( p \land (q \lor r)) \equiv ((p \land q ) \lor ( p \land r)),$$ by showing that their logic trees are identical.

Solution

You must find the logic tree for $$( p \land (q \lor r)),$$ and show that it is identical to the logic tree given in the question. The first variable to consider is $$p.$$ If $$p$$ is false, then the whole statement must be false, since it is joined by an and. If $$p$$ is true, there it is currently impossible to tell if the whole statement will be true or false, so it must branch onto a logic for $$q.$$ hence, the first branches of the logic tree will look like this: If $$p$$ is false, the whole statement must be false. If $$p$$ is true, it is unclear whether the statement is true or false yet.

Next, consider what happens when $$q$$ is true or false. If $$q$$ is true, $$q \lor r$$ must be true. Since $$p$$ is already true in this scenario, the whole statement $$( p \land (q \lor r))$$ must be true. If $$q$$ is false, it is unclear whether $$q \lor r$$ is true, and hence this must branch onto the decision for $$r.$$ The next branches of the logic tree will look like this: If $$q$$ is true, $$q \lor r$$, and hence the whole statement, must be true. If $$q$$ is false, whether or not the statement is true depends on $$r.$$

Finally, only the value of $$r$$ remains. If $$r$$ is true, $$(q \lor r)$$ must be true, meaning $$( p \land (q \lor r))$$ must be true. If $$r$$ is false, the or statement is false, meaning the whole statement must also be false since it $$p$$ is false in this scenario. Hence, the complete logic tree for $$( p \land (q \lor r))$$ will be: Given that $$p$$ is true and \9q\) is false, if $$r$$ is true, the whole statement must be true. If $$r$$ is false, the whole statement must be false.

Since this logic tree is identical to the one given in the question, so therefore the two statements must be the same.

### Results in Propositional Logic

The first important result in propositional logic are De Morgan's Laws. De Morgan's laws state that:

\begin{align} \lnot ( p \land q ) & \equiv (\lnot p \lor \lnot q) \\ \lnot (p \lor q ) & \equiv (\lnot p \land \lnot q). \end{align}

De Morgan's laws can be proven by showing that the logic trees or logic tables are the same for the right hand side and left hand side, just like the properties in the previous section.

Another important result within logic is that implies can be written just using not and or symbols.

$p \implies q \equiv \lnot p \lor q.$

This means that all the examples that you have seen so far can be written just using and, or and not connectives. Again, this can be proven by comparing logic trees or logic tables.

## Areas of Mathematical Logic Application

Mathematical logic is essential in many subfields of mathematics. Without the use of logic, many results within mathematics could never have been proven. To name a few, mathematical logic is essential in the fields of:

• Proof Theory,

• Set Theory,

• Recursion Theory,

• Model Theory.

Beyond pure mathematics, mathematical logic is essential in computer science, natural sciences, economics, and basically every other field that uses mathematics.

## Logic - Key takeaways

• A proposition is a statement that is either true or false. A connective is something that joins two propositions together, creating a new proposition. The main 4 connectives are:
• and: $$\land,$$
• or: $$\lor,$$
• not: $$\lnot,$$
• implies: $$\implies.$$
• Truth Tables and Logic Trees are used to show when more complicated propositions are true or false, based on when their individual components are true or false. Equivalencies within logic can be proven by showing that the truth tables or logic trees are identical.
• Some important properties of the connectives in propositional logic are:
• Double Negation: $\lnot (\lnot p) \equiv p.$

• Commutativity: \begin{align} p \land q & \equiv q \land p, \\ p \lor q & \equiv q \lor p. \end{align}
• Associativity: \begin{align} ( p \land (q \land r)) & \equiv ( (p \land q) \land r), \\ ( p \lor (q \lor r)) & \equiv ( ( p \lor q) \lor r). \end{align}

• Distributivity: \begin{align} ( p \land (q \lor r)) & \equiv ((p \land q ) \lor ( p \land r) ), \\ (p \lor (q \land r)) & \equiv ((p \lor q) \land (p \lor r)). \end{align}
• 2 Important properties within logic are:
• De Morgan's Laws: \begin{align} \lnot ( p \land q ) & \equiv (\lnot p \lor \lnot q), \\ \lnot (p \lor q ) & \equiv (\lnot p \land \lnot q). \end{align}
• The equivalency of implies: $p \implies q \equiv \lnot p \lor q.$

Logic is the study of the truth and how we can reach universal truths by applying mathematical deduction.

There are four parts of mathematical logic:

1. Model Theory

2. Proof Theory

3. Recursion Theory, it is also known as computability theory

4. Set theory

Direct proof is used to prove statements of the form ”if p then q” or ”p implies q” which we can write as p ⇒ q. The method of the proof is to takes an original statement p, which we assume to be true, and use it to show directly that another statement q is true.

An example of logic is deducing that two truths imply a third truth.

The Proof Theory is applied to Computer Science, AI (Artificial Intelligence), and IT (Information Technology).

## Final Logic Quiz

Question

What is a proposition?

A statement that is either true or false.

Show question

Question

What is the symbol for and?

$$\land.$$

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Question

What is the symbol for or?

$$\lor.$$

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Question

What is the symbol for not?

$$\lnot.$$

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Question

What is the symbol for implies?

$$\implies.$$

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Question

What is the property of double negation?

$$\lnot \lnot p \equiv p.$$

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Question

What is the property of commutativity of and?

$$p \land q \equiv q \land p.$$

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Question

What is the property of commutativity of or?

$$p \lor q \equiv q \lor p.$$

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Question

What is the property of associativity of and?

$$((p \land q) \land r) \equiv (p \land (q \land r)).$$

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Question

What is the property of distributivity of or over and?

$$(p \lor (q \land r)) \equiv ((p \lor q) \land (p \lor r)).$$

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Question

What are De Morgan's Laws?

\begin{align} \lnot ( p \land q ) & \equiv (\lnot p \lor \lnot q), \\ \lnot (p \lor q ) & \equiv (\lnot p \land \lnot q). \end{align}

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Question

What is implies equivalent to, in terms of and, or and not?

$$p \implies q \equiv \lnot p \lor q.$$

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Question

If $$p$$ is true and $$q$$ is true, what is $$p \land q$$?

True.

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Question

If $$p$$ is true and $$q$$ is false, what is $$p \land q$$?

True.

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Question

If $$p$$ is false and $$q$$ is true, what is $$p \land q$$?

True.

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Question

If $$p$$ is false and $$q$$ is false, what is $$p \land q$$?

True.

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Question

If $$p$$ is true and $$q$$ is true, what is $$p \lor q$$?

True.

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Question

If $$p$$ is true and $$q$$ is false, what is $$p \lor q$$?

True.

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Question

If $$p$$ is false and $$q$$ is true, what is $$p \lor q$$?

True.

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Question

If $$p$$ is false and $$q$$ is false, what is $$p \lor q$$?

True.

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Question

If $$p$$ is true and $$q$$ is true, what is $$p \implies q$$?

True.

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Question

If $$p$$ is true and $$q$$ is false, what is $$p \implies q$$?

True.

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Question

If $$p$$ is false and $$q$$ is true, what is $$p \implies q$$?

True.

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Question

If $$p$$ is false and $$q$$ is false, what is $$p \implies q$$?

True.

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