From French revolution we got equality and liberty concept.in practical life it's seems two opposite pole .How will you satisfy this argument that both can be achieved simultaneously....As liberty leads to inequality and if we transfer some ones gain to poor it will not be accepted by libertarian... Now a days there is protectionist policy around the world as in usa and eu we witnessed this contradiction very well and came to a conclusion that both can't be claimed as equality and liberty are two different thing like tom and Jerry?
1 The two can be brought together in the requirement of equal liberty for all, a principle we find in Rawls' Theory of Justice (1972).
2 To the extent that there is economic liberty - a market economy - in which free exchange takes place, there will be winners and losers and economic equality will be a vanishing quantity.
3 The flipside to this is that to the extent that economic equality is introduced and preserved, economic liberty will have to be curtailed since free exchange will produce economic inequalities.
4 Equality has many dimensions : economic equality, 'physical' equality (of age, health strength), political equality and others besides. The same holds for freedom : freedom of expression, freedom of worship, freedom of association, freedom to marry, freedom of movement, freedom from constraint or arbitrary arrest. It is not at all clear that there is any general relation between all the dimensions of equality and all the dimensions of freedom. Freedom of movement can go easily along with economic inequality; 'physical' inequality can co-exist with political freedom.
5 There are dependencies, however : political freedom can be undermined by economic inequalities. We may all have one vote but institutions with economic power may by means of massive wealth (economic inequality) exert undue political influence and subvert political freedom.
Just some connexions and tensions you might care to consider.
I agree that they oppose each other, as you can't have liberty with equality, and you can't have equality with liberty.
Equality taken as an ideal leads to communism, which is shown to fail whenever it is tried.
Liberty taken as an ideal leads to anarchy, which is mostly untested but seems unworkable.
Equality of liberty would be what I'd suggest we aim for. The same level of freedom for all, seems to make the most sense.