`\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT"`
• Let `U` be the universal set and `A` a subset of `U.` Then `\color{blue} ul(\mathtt (THE \ \ COMPLEMENT \ \ OF\ \ A))` is the set of all elements of `U` which are not the elements of `A.`
• Symbolically, we write `A′` to denote the complement of `A` with respect to `U.`
Thus, `A′ = {x : x ∈ U "and" x ∉ A }.` Obviously `A′ = U – A`
E.g. `U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}` and `A = {1, 3, 5, 7, 9}.` then
`A′ = { 2, 4, 6, 8,10 }.`
` \color{red} \ox \color{red} \mathbf(COMMON \ CONFUSION) : ` If `A` is a subset of the universal set `U`, then its complement `A′` is also a subset of `U.`
We have ` \ \ \ \ A′ = { 2, 4, 6, 8, 10 }`
Hence `(A′ )′ = {x : x ∈ U "and" x ∉ A′}`
` \ \ \ \ \ \ \ \ = {1, 3, 5, 7, 9} = A`
`\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT"`
• Let `U` be the universal set and `A` a subset of `U.` Then `\color{blue} ul(\mathtt (THE \ \ COMPLEMENT \ \ OF\ \ A))` is the set of all elements of `U` which are not the elements of `A.`
• Symbolically, we write `A′` to denote the complement of `A` with respect to `U.`
Thus, `A′ = {x : x ∈ U "and" x ∉ A }.` Obviously `A′ = U – A`
E.g. `U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}` and `A = {1, 3, 5, 7, 9}.` then
`A′ = { 2, 4, 6, 8,10 }.`
` \color{red} \ox \color{red} \mathbf(COMMON \ CONFUSION) : ` If `A` is a subset of the universal set `U`, then its complement `A′` is also a subset of `U.`
We have ` \ \ \ \ A′ = { 2, 4, 6, 8, 10 }`
Hence `(A′ )′ = {x : x ∈ U "and" x ∉ A′}`
` \ \ \ \ \ \ \ \ = {1, 3, 5, 7, 9} = A`