What is the truth value of the quantification ∀x(x² < 1) where the domain consists of (i) the positive integers greater than or equal to 1, and (ii) the integers x such that -1 < x... What is the truth value of the quantification ∀x(x² < 1) where the domain consists of (i) the positive integers greater than or equal to 1, and (ii) the integers x such that -1 < x < 1? Write the negation of the following quantified statements: (i) ∀x P(x) (ii) ∃x P(x). Express the following propositions using quantifiers and form the negation of the first three propositions: (a) There is someone in the class who cannot speak in Hindi. (b) Every child loves to eat both Pizza and Sandwich. (c) All politicians are honest. (d) Every real number except zero has a multiplicative inverse. (e) The sum of two even integers is even. Prove the following statements if true; otherwise, give counterexamples: (a) For any integers a and b, 4a + 6b + 5 is odd. (b) Product of two odd integers is odd. Read more

Steps to Solve

1. Truth value of quantification for positive integers

   For (i) the domain is positive integers ≥ 1. The expression $P(x): x^2 < 1$ is never true since $x^2$ for $x ≥ 1$ is always ≥ 1. Thus, the statement $\forall x (P(x))$ is false.

   For (ii) with integers $x$ such that $-1 < x < 1$, the only values are $0$ and numbers between $-1$ and $1$. $P(x)$ is true for $x = 0$, making the expression $\exists x (P(x))$ true.

2. Negation of quantified statements

   For (i) $\forall x P(x)$, the negation is

   $$ \exists x \neg P(x) $$

   It means there exists at least one $x$ for which $P(x)$ is false.

   For (ii) $\exists x P(x)$, the negation is

   $$ \forall x \neg P(x) $$

   It means for every $x$, $P(x)$ is false.

3. Express propositions using quantifiers

   a) "There is someone in the class who cannot speak Hindi"

   $$ \exists x , \neg S(x) $$ where $S(x)$ means "$x$ can speak Hindi".

   b) "Every child loves to eat both Pizza and Sandwich"

   $$ \forall x (C(x) \implies (P(x) \land S(x))) $$ where $C(x)$ means "$x$ is a child", $P(x)$ is "loves Pizza", and $S(x)$ is "loves Sandwich".

   c) "All politicians are honest"

   $$ \forall x (P(x) \implies H(x)) $$ where $P(x)$ is "$x$ is a politician" and $H(x)$ is "$x$ is honest".

4. Negations of the first three propositions

   a) The negation becomes

   $$ \forall x , S(x) $$

   b) Negation

   $$ \exists x (C(x) \land (\neg P(x) \lor \neg S(x))) $$

   c) Negation

   $$ \exists x (P(x) \land \neg H(x)) $$

5. Proving statements or providing counterexamples

   a) For $a, b \in \mathbb{Z}$, the statement "$4a + 6b + 5$ is odd.".

   This is true since $4a$ and $6b$ are both even. Therefore, adding 5 (odd) gives an odd result.

   b) The product of two odd integers is odd. For odd integers:

   $$ x = 2m + 1 \quad \text{and} \quad y = 2n + 1 $$

   Then $xy = (2m + 1)(2n + 1) = 4mn + 2m + 2n + 1$, which is odd.