Sets Theory (Proof with Examples)
̅
1 - Prove that 𝑨 − 𝑩 = 𝑨 ∩ 𝑩
Let us consider 𝑥 any element of the set 𝐴 − 𝐵
⟹𝑥 ∈𝐴−𝐵
⟹ 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∉ 𝐵
⟹ 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵̅
⟹ 𝑥 ∈ 𝐴 ∩ 𝐵̅
̅̅̅̅̅̅̅̅
2 - Prove that 𝑨 ∪𝑩=𝑨 ̅∩𝑩
̅
Let us consider 𝑥 any element of the set ̅̅̅̅̅̅̅
𝐴∪𝐵
⟹ 𝑥 ∈ ̅̅̅̅̅̅̅
𝐴∪𝐵
⟹𝑥 ∉ 𝐴∪𝐵
⟹ 𝑥 ∉ 𝐴 𝑎𝑛𝑑 𝑥 ∉ 𝐵
⟹ 𝑥 ∈ 𝐴̅ 𝑎𝑛𝑑 𝑥 ∈ 𝐵̅
⟹ 𝑥 ∈ 𝐴̅ ∩ 𝐵̅
̅̅̅̅̅̅̅̅
3 - Prove that 𝑨 ∩𝑩=𝑨 ̅∪𝑩
̅
Let us consider 𝑥 any element of the set ̅̅̅̅̅̅̅̅
𝑨∩𝑩
⟹ 𝑥 ∈ ̅̅̅̅̅̅̅
𝐴∩𝐵
⟹𝑥 ∉ 𝐴∩𝐵
⟹ 𝑥 ∉ 𝐴 𝑜𝑟 𝑥 ∉ 𝐵
⟹ 𝑥 ∈ 𝐴̅ 𝑜𝑟 𝑥 ∈ 𝐵̅
⟹ 𝑥 ∈ 𝐴̅ ∪ 𝐵̅
̅
1 - Prove that 𝑨 − 𝑩 = 𝑨 ∩ 𝑩
Let us consider 𝑥 any element of the set 𝐴 − 𝐵
⟹𝑥 ∈𝐴−𝐵
⟹ 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∉ 𝐵
⟹ 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵̅
⟹ 𝑥 ∈ 𝐴 ∩ 𝐵̅
̅̅̅̅̅̅̅̅
2 - Prove that 𝑨 ∪𝑩=𝑨 ̅∩𝑩
̅
Let us consider 𝑥 any element of the set ̅̅̅̅̅̅̅
𝐴∪𝐵
⟹ 𝑥 ∈ ̅̅̅̅̅̅̅
𝐴∪𝐵
⟹𝑥 ∉ 𝐴∪𝐵
⟹ 𝑥 ∉ 𝐴 𝑎𝑛𝑑 𝑥 ∉ 𝐵
⟹ 𝑥 ∈ 𝐴̅ 𝑎𝑛𝑑 𝑥 ∈ 𝐵̅
⟹ 𝑥 ∈ 𝐴̅ ∩ 𝐵̅
̅̅̅̅̅̅̅̅
3 - Prove that 𝑨 ∩𝑩=𝑨 ̅∪𝑩
̅
Let us consider 𝑥 any element of the set ̅̅̅̅̅̅̅̅
𝑨∩𝑩
⟹ 𝑥 ∈ ̅̅̅̅̅̅̅
𝐴∩𝐵
⟹𝑥 ∉ 𝐴∩𝐵
⟹ 𝑥 ∉ 𝐴 𝑜𝑟 𝑥 ∉ 𝐵
⟹ 𝑥 ∈ 𝐴̅ 𝑜𝑟 𝑥 ∈ 𝐵̅
⟹ 𝑥 ∈ 𝐴̅ ∪ 𝐵̅
