# Math Help - De morgans methods; simple questions, what do they mean in normal English?

1. ## De morgans methods; simple questions, what do they mean in normal English?

Hi;

I'm trying to figure out if the following statements are true or not;

1.
p v ¬q = p -> q

2. ¬q -> ¬p = q -> p

3. p v (q -> r) = (p v q) -> (p v r)

Can someone please outline what these lines mean in normal english?

For example, the start of 1 means; p or not q...

I don't know what that little arrow actually means...

Are there any resources online to learn this at basic level?

2. Originally Posted by Student122
Hi;

I'm trying to figure out if the following statements are true or not;

1. p v ¬q = p -> q

2. ¬q -> ¬p = q -> p

3. p v (q -> r) = (p v q) -> (p v r)

Can someone please outline what these lines mean in normal english?

For example, the start of 1 means; p or not q...

I don't know what that little arrow actually means...

Are there any resources online to learn this at basic level?

Hi Student122,

The little arrow means "implies". $p \rightarrow q$ means p implies q, or simply "if p, then q".

Sometimes it's written $p \implies q$

Do you know what truth table are? Look here: Truth table - Wikipedia, the free encyclopedia

Code:

p | q |not p | not q | p --> q | not p or q | p or not q
---------------------------------------------------------
T | T | F    |  F    |    T    |       T    |     T
T | F | F    |  T    |    F    |       F    |     T
F | T | T    |  F    |    T    |       T    |     F
F | F | T    |  T    |    T    |       T    |     T
As you can see from the truth table above, your first implication is false.

[1] $p \vee \neg q \equiv p \rightarrow q$ is false.

Their truth values are not the same. Check the truth table.

But this is true: $\neg p \vee q \equiv p \rightarrow q$.

Chech the truth values under their columns.

Try to set up a truth table to verify the other 2.

3. Originally Posted by masters
Hi Student122,

The little arrow means "implies". $p \rightarrow q$ means p implies q, or simply "if p, then q".

Sometimes it's written $p \implies q$

Do you know what truth table are? Look here: Truth table - Wikipedia, the free encyclopedia

Code:

p | q |not p | not q | p --> q | not p or q | p or not q
---------------------------------------------------------
T | T | F    |  F    |    T    |       T    |     T
T | F | F    |  T    |    F    |       F    |     T
F | T | T    |  F    |    T    |       T    |     F
F | F | T    |  T    |    T    |       T    |     T
As you can see from the truth table above, your first implication is false.

[1] $p \vee \neg q \equiv p \rightarrow q$ is false.

Their truth values are not the same. Check the truth table.

But this is true: $\neg p \vee q \equiv p \rightarrow q$.

Chech the truth values under their columns.

Try to set up a truth table to verify the other 2.
Hi masters,

Can I attempt to 'translate' the first question?

I think it's saying;

p or not q = p implies q

So looking at the table, for the first part we look at the last column;

p | q |not p | not q | p --> q | not p or q | p or not q
---------------------------------------------------------
T | T | F | F | T | T | T
T | F | F | T | F | F | T
F | T | T | F | T | T | F
F | F | T | T | T | T | T

For the last part of the equation we look at column 5 (in bold & red)

Correct me if I'm wrong, I probably am (but don't know why lol), but aren't they both True and doesn't that mean that the original sum is true and not false?

4. Originally Posted by Student122
Hi masters,

Can I attempt to 'translate' the first question?

I think it's saying;

p or not q = p implies q

So looking at the table, for the first part we look at the last column;

p | q |not p | not q | p --> q | not p or q | p or not q
---------------------------------------------------------
T | T | F | F | T | T | T
T | F | F | T | F | F | T
F | T | T | F | T | T | F
F | F | T | T | T | T | T

For the last part of the equation we look at column 5 (in bold & red)

Correct me if I'm wrong, I probably am (but don't know why lol), but aren't they both True and doesn't that mean that the original sum is true and not false?
All truth values in the columns must match, not just the first one.

The red columns don't match.

5. Originally Posted by masters
All truth values in the columns must match, not just the first one.

The red columns don't match.