Sets and maps

101. Let A_i (i ∈ I), B be subsets of X. Prove that:

( ∪ i ∈ I A i )     ∩   B = ∪ i ∈ I ( A i ∩ B ) ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756454/84bc859e-388a-4821-81f7-7fb447dc7024/content/eq1.tif"/>

( ∩ i ∈ I A i )     ∪   B = ∩ i ∈ I ( A i ∪ B ) ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756454/84bc859e-388a-4821-81f7-7fb447dc7024/content/eq2.tif"/>

∪ i ∈ I A i ¯ = ∩ i ∈ I A i ¯ ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756454/84bc859e-388a-4821-81f7-7fb447dc7024/content/eq3.tif"/>

∩ i ∈ I A i ¯ = ∪ i ∈ I A i ¯ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756454/84bc859e-388a-4821-81f7-7fb447dc7024/content/eq4.tif"/>