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Q.1Are the following statements True or False? (briefly justify your answer) (a...

Mar 15, 2024

Q.1Are the following statements True or False? (briefly justify your answer) (a) (i)

\emptyset=\{0\}

(ii)

x \in\{x\}

(iii)

\emptyset=\{\emptyset\}

(iv)

\emptyset \in\{\emptyset\}

(b) The following sets have been expressed in set builder notation. Re-express them in list notation. (i)

\{\rho: \rho

is a state capital,

\rho

is in Australia

\}

(ii)

\{x: x=2 n-5

where

n \in \mathbb{N}\}

(iii)

\left\{y: 2 y^{2}=50\right.

where

\left.y \in \mathbb{Z}\right\}

4 marks (iv)

\left\{z: 3 z=n^{2}\right.

where both

n

and

z

are natural numbers

\}

Q. 2 On the set

A=\{1,2,3,4,5\}

we mav define two functions

f: A \longrightarrow A

and

g: A \longrightarrow A

by the

Solution by Steps

step 1

To determine the truth value of the statement

\emptyset = \{0\}

, we compare the empty set to a set containing the number 0

step 2

The empty set has no elements, while the set

\{0\}

has one element, 0. Therefore,

\emptyset \neq \{0\}

and the statement is False

step 3

To determine the truth value of the statement

x \in \{x\}

, we consider any element x and whether it is a member of the set containing only x itself

step 4

Since x is the only element in the set

\{x\}

, it is true that

x \in \{x\}

for any x. Therefore, the statement is True

step 5

To determine the truth value of the statement

\emptyset = \{\emptyset\}

, we compare the empty set to a set containing the empty set

step 6

The empty set has no elements, while the set

\{\emptyset\}

has one element, which is the empty set itself. Therefore,

\emptyset \neq \{\emptyset\}

and the statement is False

step 7

To determine the truth value of the statement

\emptyset \in \{\emptyset\}

, we check if the empty set is an element of the set containing the empty set

step 8

Since the set

\{\emptyset\}

contains the empty set as its only element, it is true that

\emptyset \in \{\emptyset\}

and the statement is True

step 9

To re-express the set builder notation

\{\rho: \rho \text{ is a state capital, } \rho \text{ is in Australia }\}

in list notation, we list the state capitals of Australia

step 10

The state capitals in Australia are Sydney, Melbourne, Brisbane, Perth, Adelaide, Hobart, Darwin, and Canberra. Therefore, the list notation is

\{Sydney, Melbourne, Brisbane, Perth, Adelaide, Hobart, Darwin, Canberra\}

step 11

To re-express the set builder notation

\{x: x = 2n - 5 \text{ where } n \in \mathbb{N}\}

in list notation, we generate values of x for natural numbers n

step 12

The first few natural numbers are 1, 2, 3, ... . Substituting these into the expression

2n - 5

, we get -3, -1, 1, 3, ... . Therefore, the list notation is

\{-3, -1, 1, 3, ...\}

step 13

To re-express the set builder notation

\{y: 2y^2 = 50 \text{ where } y \in \mathbb{Z}\}

in list notation, we solve the equation for y

step 14

Solving

2y^2 = 50

gives

y^2 = 25

, so

y = \pm5

. Since y must be an integer, the list notation is

\{-5, 5\}

step 15

To re-express the set builder notation

\{z: 3z = n^2 \text{ where both } n \text{ and } z \text{ are natural numbers }\}

in list notation, we find values of z that satisfy the equation for natural numbers n

step 16

The first few natural numbers are 1, 2, 3, ... . Squaring these and dividing by 3, we get z values of 1/3, 4/3, 3, ... . However, z must be a natural number, so the only valid z value from these is 3. Therefore, the list notation is

\{3\}

Answer

(a)(i) False, (ii) True, (iii) False, (iv) True (b)(i)

\{Sydney, Melbourne, Brisbane, Perth, Adelaide, Hobart, Darwin, Canberra\}

(ii)

\{-3, -1, 1, 3, ...\}

(iii)

\{-5, 5\}

(iv)

\{3\}

Key Concept

Set theory and set builder notation

Explanation

The truth value of statements involving sets is determined by the definition of the empty set and membership within sets. Set builder notation is a concise way to define a set by specifying a property that its members must satisfy, and it can be converted to list notation by finding all elements that meet the given property.