Define the terms on the basis of oxidation number concept: OxidationReductionReducing agentOxidizing agent

In terms of Classical idea:

Oxidation: Addition of  oxygen or any other electronegative element to a substance .Conversely , Removal of Hydrogen or any other electropositive element from a substance. 

Mg (s) + S (s) MgS (s)

Similarly, Reduction  is just opposite process of oxidation.

2FeCl₃(aq) + H₂(g)2FeCl₂(aq) + 2HCl (aq)

In terms of Electron transfer reaction:
Oxidation: Loss of electron 
Reduction: Gain of electron 
Oxidant: Acceptor of electron
...

On the basis of classical concepts, definitions of following terms are:

• Oxidation: The addition of oxygen or electronegative radical/element; or removal or hydrogen or electropositive element/radical is said to be oxidation.
• Reduction: The addition of hydrogen or electropositive radical/element or removal or oxygen or electronegative element/radical is said to be reduction.
• Reducing agent: A substance that experiences increase in Oxidation number of one or more of its elements is said to...

Let the oxidation number of each element to be found be x in each cases.

Here, ON of K, H = 1; ON of O = -2

a) KMnO₄

It is a molecule hence, the ON of the molecule is zero.

1 + x + 4(-2) = 0

1 + x - 8 = 0

x - 7 = 0

x = 7

b) K₂Cr₂O₇

It is a molecule hence, the ON of the molecule is zero.

2(1) + 2(x) + 7(-2) = 0

1 + x - 7 = 0

x - 6 = 0

x = 6

c) H₂SO₄

It is a molecule hence, the ON of the molecule is zero.

2(1) + x + 4(-2) = 0

2 + x - 8 = 0

x - 6 = 0

x = 6

d) CO₃^2-

It is a radical that has -2 charge so,...

let it be x. from structure , it's 4 O-atoms are peroxide. 

x + (-2x 1 ) + ( -1 x 4 ) = 0

thus, X = + 6