How to represent proposition symbolically? (discrete math)?

• February 16th 2010, 01:32 AM
bugbear
How to represent proposition symbolically? (discrete math)?
p : You run 10 laps daily
q : You are healthy.
r : You take multi vitamins

1. you run 10 laps daily, but you are not healthy.
2. you run 10 laps daily, you take multi-vitamins, and you are healthy.
3. you run 10 laps daily or you take multi-vitamins, and you are healthy.

would appreciate if u guys can help.
• February 16th 2010, 01:58 AM
bmp05
hey bugbear,
at the moment, you just have to connect the propositions with logical "and" or "or", there's no implication or anything else, so for example:

1. $p \wedge \neg q$

"p and (aka. but) not q"
• February 16th 2010, 02:03 AM
bugbear
oh, thank you for clearing that.
• February 16th 2010, 06:22 AM
bugbear
one more question, how to determine whether the sentence is proposition? and if its a proposition, how to write its negation?

for example this sentence,

Code:

`The difference of two primes.`
• February 16th 2010, 07:13 AM
bmp05
hello bugbear,
a proposition can be evaluated as being either true or false. So for example:
This a proposition because it is either true, and your username is bugbear, or it's false and you have some other username.

An example of a sentence which isn't a proposition is:
Because this sentence cannot be answered with true or false.

Another example of a sentence which isn't a proposition and therefore isn't a statement is:
"The yellow cat."
A bit like your example this isn't even a proper sentence, because there's no verb.

"The yellow cat is sick," on the other hand would be a proposition.
• February 16th 2010, 07:20 AM
Hello bugbear
Quote:

Originally Posted by bugbear
one more question, how to determine whether the sentence is proposition? and if its a proposition, how to write its negation?

for example this sentence,

Code:

`The difference of two primes.`

A sentence is a proposition if it has a truth value; in other words, it is a statement (not a question, not a command, ...) that is either true or false.

So obviously
The difference of two primes
is not a proposition, because it isn't a statement. In fact, it isn't even a sentence.

Examples of propositions are:
All cows have six legs.

The Pacific Ocean is the deepest ocean in the world.

Every mathematics lecturer is good-looking.
(The first is clearly false, the second is true. For the third, it's only a proposition provided the term 'good-looking' has been unambiguously defined.)

But examples of sentences that aren't propositions include:
Eat more chocolate!

Is 6 greater than 2?

Why is logic so hard?
(Note that, although the second of these can be answered by 'Yes', it doesn't make it a proposition: it's a question.)

To write the negation of a proposition, you can always write in front of it the phrase:
It is not true that ...
Try it with the three propositions above. You'll see that each one now has the opposite truth value.

(By the way, have you now solved your original questions 2 and 3?)