Answer to Question #344353 in Discrete Mathematics for Ttypre

Question

1. Express the following statements using quantifiers.

A. All cats have fleas.

B. No one in this class knows how to speak Mandarin.

2.Form the negation of each statement without applying negation to the left of any quantifier.

3. Express each negation in simple English.

Part 2

1. Given A = {4, 8, 12, 16} and B = {6, 12, 18, 24}, determine the following:

a. A U B

b. A ∩ B

c. A - B

d. B - A

Answer 1

1.

A. All cats have fleas.

Domain: cats

P(x): x has fleas

$\forall x (P(x))$

B. No one in this class knows how to speak Mandarin.

Domain: people in this class

P(x): x knows how to speak Mandarin

$\lnot \exist x (P(x)) = \forall x (\lnot P(x))$

2. Negation:

A. $\lnot (\forall x (P(x))) = \exist x (\lnot P(x))$

B. $\lnot ( \forall x (\lnot P(x))) = \exist x (P(x))$

3. In simple English:

A. There is a cat which doesn’t have any fleas.

B. Someone in this class Knows how to speak Mandarin.

Part 2.

A = {4, 8, 12, 16}, B = {6, 12, 18, 24}.

a. A U B = {4, 6, 8, 12, 16, 18, 24}

(A U B = { $x : x \in A$ or $x \isin B$ })

b. A ∩ B = {12}

(A ∩ B = { $x : x \in A$ and $x \isin B$ })

c. A - B = {4, 8, 16}

(A - B = { $x : x \in A$ and $x \notin B$ })

d. B - A = {6, 18, 24}

(B - A = { $x : x \in B$ and $x \notin A$ })