Quantifier

Q. If $A=\{1,2,3,4\}.$, then number of statements from the below having truth value as true is
$\left(1\right)\exists x\in A$ such that $x+3=8$
$\left(2\right)\forall x\in A,x+2<7.$
$\left(3\right)\forall x\in A$ such that $x+1<3.$
$\left(4\right)\forall x\in A,x+3\ge 5.$

Q. If $A=\{1,2,3,4\}$, then number of statements from the below having truth value as true is
$\left(1\right)\exists x\in A$ such that $x+3=8$
$\left(2\right)\forall x\in A,x+2<7.$
$\left(3\right)\forall x\in A$ such that $x+1<3.$
$\left(4\right)\forall x\in A,x+3\ge 5.$

Q. If a then b and if b then c $⟹$ If a then c.

1. False
2. True

Q. If $x$ satisfies $|x-1|+|x-2|+|x-3|\ge 6$, then

1. $0\le x\le 4$
2. $x\le -2$ or $x\ge 4$
3. $x\le 0$ or $x\ge 4$
4. $x\le 0$ or $x\ge 2$

Q. $\mathrm{If}Q:\mathrm{Set}\mathrm{of}\mathrm{Rational}\mathrm{numbers}N:\mathrm{Set}\mathrm{of}\mathrm{Natural}\mathrm{numbers}\mathrm{Then}"\forall q\in Q\exists n\in N\mathrm{such}\mathrm{that}q<n.",then-$

1. For any rational number q there exists a natural number n such that q<n.
2. There is a natural number which is greater than every rational number.
3. Every rational number is less than natural number.
4. There exists natural and rational number(n and q) such that q<n

Q. $\mathrm{What}\mathrm{can}\mathrm{be}\mathrm{said}\mathrm{about}\mathrm{the}\mathrm{following}\mathrm{statements}?\mathrm{Statement}1:\exists a\mathrm{natural}\mathrm{number}a\mathrm{such}\mathrm{that}{a}^2\le a.\mathrm{Statement}2:\forall \mathrm{rational}\mathrm{numbers}a,a=\frac{p}{q},\mathrm{where}p\mathrm{and}q\mathrm{are}\mathrm{integers}\mathrm{and}q\ne 0.$

1. Statement 1 is true , Statement 2 is false.
2. Statement 1 is false , Statement 2 is true.
3. Statement 1 and 2 both are true.
4. Statement 1 and 2 both are false.

Q.

If $S$ be a non-empty subset of $R$. Consider the following statement $P$: There is a rational number $x\in S$ such that $x>0$. Which of the following statements is the negation of statement $P$$?$

1. There is a rational number $x\in S$ such that $x\le 0$

2. There is no rational number $x\in S$ such that $x\le 0$

3. Every rational number $x\in S$ such that $x\le 0$

4. $x\in S\text{and}x\le 0\Rightarrow x$ is not rational

Q.

If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?

Q.

Are the following pairs of statements negations of each other?

(i) The number x is not a rational number.

The number x is not an irrational number.

(ii) The number x is a rational number.

The number x is an irrational number.

Q.

what does set Q means in relations and functions?