Key points chapter 1 book 1

If  a set satisfies :

Closure property then it is called groupoid.

Closure + associative then it is called semi-group.

Closure+ Associative+ identity element then it is called monoid.

Closure+ Associative+ identity + inverse then it is called group.

If a set satisfies all the 11 properties then it is called field

Set of real numbers, rational numbers and complex numbers are field.

Note:

Identity element in a group is always unique.

Inverse of an element is always unique.

Total number of binary operations are 4.

Powers of i :

If you have to tackle an MCQ which has high powers of  then follow the following points:

1.     Divide the power by 4.

2.     If remainder is 0 then answer is +1.

3.     If remainder is 2 then answer is  -1.

4.     If remainder is 1 then answer is +i .

5.     If remainder is 3 then answer is   -i .

Sets:

Let n(A) = n     (let number of elements in set A is n)

2ⁿ = number of all possible subsets (or power sets)

2ⁿ-1=2ⁿ-2⁰= number of all proper subsets

2ⁿ-1= number of non-empty subsets.

And

2ⁿ-2 =2( 2^n-1-1) non-empty proper subsets.

Note:

Power set has only one improper subset.

Number of elements in a set= order of set(cardinality)

{ } this is null set.

{0} this is singleton set.

In singleton set:

All subsets are 2. 1 proper and 1 improper.

A set having no proper subset is empty set.