Transfinite Cardinal Numbers

For transfinite cardinal number α, using Zorn’s lemma we have given a simple proof which is understandable to undergraduate students, of the result α + α = αα = α, that is, idempotency for addition and multiplication. Moreover for a cardinal number  with 2 ≤  < α we obtain easily α +  = α = α, α  = α α = 2 α , α  < . Using these results we get many results directly as א0 + ∁ = א0∁ = ∁ + ∁ = ∁∁ = ∁, א0 א0 = ∁ = ∁0 = 20 , ∁ = א0 ∁ = 2 where א0 = card N, ∁ = card R. -------------------------------------------------------------------------------------------------------------------------------------Date of Submission: 05-08-2018 Date of acceptance: 22-08-2018 -------------------------------------------------------------------------------------------------------------------------------------