Discrete Maths problem

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bushra1175

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Hi there. I wasn't sure where to post this so I thought here would be best.

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I know what each of them reads:

a) p is true and q is true or r is not true
b) (if p is true then q is not true) and p is not true

I'm not sure how I would go about negating these formulae. My guess for the first one is below but I'm not sure what to do about the ∨ and ∧ signs. Are they flipped when negating a formula?

1a) ¬p∧¬q∨r

As for the second one, I don't have the slightest clue where to begin. I've tried to use de morgan's law but I can't see how it applies to either of these formulae. If I could see the steps taken to reach the answer for one of them, that would help me immensely in understanding how to work out similar problems. Thanks.
 
bushra1175 said:
Hi there. I wasn't sure where to post this so I thought here would be best.

View attachment 24006

I know what each of them reads:

a) p is true and q is true or r is not true
b) (if p is true then q is not true) and p is not true

I'm not sure how I would go about negating these formulae. My guess for the first one is below but I'm not sure what to do about the ∨ and ∧ signs. Are they flipped when negating a formula?

1a) ¬p∧¬q∨r

As for the second one, I don't have the slightest clue where to begin. I've tried to use de morgan's law but I can't see how it applies to either of these formulae. If I could see the steps taken to reach the answer for one of them, that would help me immensely in understanding how to work out similar problems. Thanks.
Click to expand...
I would prefer that the first had a set of parentheses in it. But as luck would have it your answer is correct.
For #2 you need to know that \(\neg(A\to B)\equiv (A\wedge\neg B)\)
 
I hate to disagree with pka especially when it comes to set theory but I would say that when you say part a reads p is true and q is true or r is not true you are wrong. It should be p is true OR q is true AND r is not true.

One way to answer how to negate something is to say that it is not the case that p is true OR q is true AND r is not true, ie ¬[p is true OR q is true AND r is not true]

Alternatively you can think of p is true OR q is true AND r is not true as [p is true OR q is true] AND [r is not true] and negate this which should be simple.
 
Jomo said:
I hate to disagree with pka especially when it comes to set theory but I would say that when you say part a reads p is true and q is true or r is not true you are wrong. It should be p is true OR q is true AND r is not true.
Click to expand...
I did say that the lack of parentheses was a problem. If one is a student of Copi the order of preference is \(\bf\equiv~\to~\vee~\wedge~\& ~\neg\) .
So we should read 1.a as \(p\vee(q\wedge\neg r)\) So that its negation would be \(\neg p\wedge (\neg q\vee r)\)
Which according to Copi is \(\neg p\wedge \neg q\vee r\)
 
@Jomo I agree with pka that parentheses would help, and I agree with you that the OP’s translation of the first problem into English is wrong. but I do not think pka was talking about the OP’s translation into English. I suspect that pka merely meant that

[MATH]\neg \{(p \lor q) \land \neg r\} ]= (\{\neg (p \lor q)\} \lor \{\neg(\neg r)\}) = \{(\neg p \land \neg q) \lor r\}.[/MATH]
Somehow or other, the OP came up with the correct answer.
 
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pka said:
I would prefer that the first had a set of parentheses in it. But as luck would have it your answer is correct.
For #2 you need to know that \(\neg(A\to B)\equiv (A\wedge\neg B)\)
Click to expand...

Thanks for this. I answered question (a) by simply guessing that the ∨ and ∧ signs are flipped. Does this mean they're always flipped when negating the formula?
 
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pka said:
I would prefer that the first had a set of parentheses in it. But as luck would have it your answer is correct.
For #2 you need to know that \(\neg(A\to B)\equiv (A\wedge\neg B)\)
Click to expand...

Also, I applied the equivalence you gave me to the second question and I ended up with this

¬ ((p → ¬q) → p)

Is this correct?
 
Let's try this concretely.

Either my pet is a cat or a dog.

How do I express the negation of that. Consider this possibility.

Either my pet is not a cat or is not a dog.

That will be true if my pet is a cat. The "my pet is not a cat" is false, but the "not a dog" is true so the compound statement is true.

That will also be true if my pet is a dog. The "not cat" part is true even though the "not dog" part is false.

So a true negation will be

My pet is not a cat and not a dog.

Let's say that x is a female US senator. Then, this is a true statement:

x is a woman and a US senator.

Now suppose that x is not a female US senator. Then the statement in the preceding line is false, and its negation will be true. Consider this proposed statement.

x is not a woman and not a US senator.

But if x is a woman who is not a US senator that statement will be false. Similarly, if x is a male US senator, that statement will be false. So it is possible for the proposed negation to be false when the original statement is false. Therefore, it is not a proper negation of the original statement.

x is not a woman or not a US Senator.

That statement will be true for any woman who is not a US senator. It will also be true for any man who is not a US senator. The only time it will be false is if x is a female US senator. Thus it does negate the claim that x is a female US senator.
 
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