ALLAMA IQBAL OPEN UNIVERSITY
ISLAMABAD
Pre- Calculus (4431)
SUBMITTED BY: AHMED HUSSAIN SHAH
STUDENT I’D: 0000512568
SEMESTER: Autumn, 2024
ASSIGNMENT NO. 1
,QUESTION NO.1
(a) Which of the following sets have closure property
with respect to addition and multiplication;
i) {1} ii) {0, –1} iii) {1, –1}
(b) Prove is an irrational number.
Closure Property and Irrational Numbers
Closure Property
The closure property of a set with respect to a binary operation (such as
addition or multiplication) states that the result of combining any two
elements of the set using that operation will always be an element of the
same set.
Sets with Closure Property
Let’s examine the given sets:
i){1}: This set has closure property with respect to multiplication, since 1
× 1 = 1, which is an element of the set. However, it does not have
closure property with respect to addition, since 1 + 1 = 2, which is not an
element of the set.
, ii){0, –1}: This set has closure property with respect to multiplication,
since 0 × 0 = 0, 0 × –1 = 0, and –1 × –1 = 1 (wait, 1 is not an element of
this set!). So, it does not have closure property with respect to
multiplication. However, it does have closure property with respect to
addition, since 0 + 0 = 0, 0 + –1 = –1, and –1 + –1 = –2 (no, –2 is not an
element of this set!). So, it does not have closure property with respect
to addition.
iii) {1, –1}: This set has closure property with respect to multiplication,
since 1 × 1 = 1, 1 × –1 = –1, and –1 × –1 = 1, all of which are elements
of the set. However, it does not have closure property with respect to
addition, since 1 + 1 = 2, which is not an element of the set.
(b) Prove is an irrational number.
Irrational Numbers
An irrational number is a real number that cannot be expressed as a
finite decimal or fraction. In other words, it is a number that cannot be
written in the form a/b, where a and b are integers and b is non-zero.
Proof that √2 is an Irrational Number
Let‟s assume that √2 is a rational number, which means it can be
expressed as a fraction a/b, where a and b are integers and b is non-zero.
ISLAMABAD
Pre- Calculus (4431)
SUBMITTED BY: AHMED HUSSAIN SHAH
STUDENT I’D: 0000512568
SEMESTER: Autumn, 2024
ASSIGNMENT NO. 1
,QUESTION NO.1
(a) Which of the following sets have closure property
with respect to addition and multiplication;
i) {1} ii) {0, –1} iii) {1, –1}
(b) Prove is an irrational number.
Closure Property and Irrational Numbers
Closure Property
The closure property of a set with respect to a binary operation (such as
addition or multiplication) states that the result of combining any two
elements of the set using that operation will always be an element of the
same set.
Sets with Closure Property
Let’s examine the given sets:
i){1}: This set has closure property with respect to multiplication, since 1
× 1 = 1, which is an element of the set. However, it does not have
closure property with respect to addition, since 1 + 1 = 2, which is not an
element of the set.
, ii){0, –1}: This set has closure property with respect to multiplication,
since 0 × 0 = 0, 0 × –1 = 0, and –1 × –1 = 1 (wait, 1 is not an element of
this set!). So, it does not have closure property with respect to
multiplication. However, it does have closure property with respect to
addition, since 0 + 0 = 0, 0 + –1 = –1, and –1 + –1 = –2 (no, –2 is not an
element of this set!). So, it does not have closure property with respect
to addition.
iii) {1, –1}: This set has closure property with respect to multiplication,
since 1 × 1 = 1, 1 × –1 = –1, and –1 × –1 = 1, all of which are elements
of the set. However, it does not have closure property with respect to
addition, since 1 + 1 = 2, which is not an element of the set.
(b) Prove is an irrational number.
Irrational Numbers
An irrational number is a real number that cannot be expressed as a
finite decimal or fraction. In other words, it is a number that cannot be
written in the form a/b, where a and b are integers and b is non-zero.
Proof that √2 is an Irrational Number
Let‟s assume that √2 is a rational number, which means it can be
expressed as a fraction a/b, where a and b are integers and b is non-zero.
