Need help in negation of statements

• issacnewton
In summary, the conversation discusses how to negate a series of statements and then translate them back into English. It covers various scenarios, such as everyone having a friend who needs help with their homework, everyone having a roommate who dislikes everyone, and someone in the freshman class not having a roommate. It also explores the idea of everyone liking someone but no one liking everyone. The summary also includes the translation of these statements into logical notation and back into English.
issacnewton
Hi

I have to negate the following statements and then express again
in English. I need to know if I am making any mistakes.

a)Everyone who is majoring in math has a friend who needs
help with his homework.

b)Everyone has a roommate who dislikes everyone.

c)There is someone in the freshman class who doesn't
have a roommate.

d)Everyone likes someone,but no one likes everyone.

a) Let P(x)= x is majoring in math
Q(x)=x needs help with homework
M(x,y)=x and y are friends

The statement would be

$$\exists x (P(x))\rightarrow \exists y (M(x,y)\wedge Q(y))$$

So the negated statement would be

$$\neg \left[\exists x (P(x))\rightarrow \exists y (M(x,y)\wedge Q(y))\right]$$

$$\neg \left[\neg(\exists x (P(x)) \vee \exists y (M(x,y)\wedge Q(y))\right]$$

$$\neg \left[\forall x \neg P(x) \vee \exists y (M(x,y)\wedge Q(y))\right]$$

$$\neg (\forall x \neg P(x)) \wedge \neg \exists y (M(x,y)\wedge Q(y))$$

$$\exists x P(x) \wedge \forall y \neg (M(x,y)\wedge Q(y))$$

$$\exists x P(x) \wedge \forall y (\neg M(x,y) \vee \neg Q(y))$$

To translate this statement back to English would be

Some person is majoring in math and everyone is either not a
friend of this person or doesn't need help in homework.

b) let Q(x,y)=x and y are roommates
M(x,y)=x likes y

The statement would be

$$\forall x \left[\exists y (Q(x,y) \wedge \forall z(\neg M(y,z)))\right]$$

so the nagated statement would be

$$\neg \forall x \left[\exists y (Q(x,y)\wedge \forall z(\neg M(y,z)))\right]$$

$$\exists x \forall y \left[\neg Q(x,y) \vee \exists z(M(y,z)) \right]$$

Translation:

Either there is some person who is not roommate with anybody or there is
someone who is liked by all.

c)let P(x)= x is in freshman class.
M(x)=x has a roommate.

The statement would be

$$\exists x \left[ P(x)\wedge \neg M(x) \right]$$

So the negated statement is

$$\neg \exists x \left[ P(x)\wedge \neg M(x) \right]$$

$$\forall x \left[ \neg P(x) \vee M(x) \right]$$

Translation:

Everyone either is not in freshman class or has a roommate.

d)let M(x,y)= x likes y

The statement would be

$$\forall x \left[ \exists y (M(x,y))\wedge \exists z \neg M(x,z) \right]$$

The negated statement would be

$$\neg \forall x \left[ \exists y (M(x,y))\wedge \exists z \neg M(x,z) \right]$$

$$\neg \left[\forall x \exists y (M(x,y))\wedge \forall x \exists z \neg M(x,z) \right]$$

$$(\neg \forall x \exists y M(x,y)) \vee (\neg \forall x \exists z \neg M(x,z) )$$

$$\left[\exists x \forall y \neg M(x,y)\right] \vee \left[\exists x \forall z M(x,z)\right]$$

Translation:

Either there is someone who likes everyone or there is someone who doesn't like
everyone.

Thanks

any help ?

1. What is negation in logic?

Negation in logic is the process of expressing the opposite or negative form of a statement. It is denoted by the symbol "~" or "not".

2. Why is negation important in scientific research?

Negation allows scientists to consider alternative explanations or hypotheses for their findings. It helps to critically evaluate and refine theories and conclusions.

3. How do you represent negation in a scientific statement?

Negation can be represented in a scientific statement by using the words "not", "never", "cannot", or by using the symbol "~" or "not". For example, "The experiment did not support the hypothesis." or "The results showed that the hypothesis was ~correct."

4. What is the difference between negation and contradiction?

Negation is the opposite or denial of a statement, while contradiction is a statement that is opposite or conflicting with another statement. Negation can be used to express a contradiction, but not all contradictions are expressed through negation.

5. How can negation be used to prove a hypothesis?

Negation can be used to test the validity of a hypothesis. If the negation of a hypothesis is true, then the original hypothesis must be false. This can help scientists to refine their theories and draw more accurate conclusions from their research.

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