Mathematics Difference of sets

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Difference of sets

The difference of the sets `A` and `B` in this order is the set of elements which belong to `A` but not to `B`. Symbolically, we write `A - B` and read as `text( A minus B.)`
Using the set builder notation, we can rewrite the definition of difference as `A - B = { x : x ∈ A and x ∉ B }`
The difference of two sets `A` and `B` can be represented by Venn diagram as shown in Fig The shaded portion represents the difference of the two sets `A` and `B.`

`text(Note :)`

`1. A-B != B-A`
2. The sets `A - B, B- A` and `A nn B` are disjoint sets.
3. `A - B ⊆ A` and `B - A ⊆ B`
4. `A - phi = A` and `A - A = phi`

For example,
If `A= {1, 2, 3, 4} and B == {4, 5, 6, 7}, then A-B = {1, 2, 3}.`

Symmetric Difference of Two Sets

Let A and B be two sets. The symmetric difference of sets A and B is the Bet
`(A- B) uu (B - A)` or `(A uu B) - (A nn B)` and is denoted by `A Delta B` or `A oplus B`
(A direct sum B).
I.e., `A oplus B` or `A Delta B = (A- B) uu (B- A)`
and `A oplus B` or `A Delta B = (A uu B) - (B nn A)`

`text(Note)`

`1. A Delta B = {x: x in A` and `x notin B}`
or `A Delta B = {x: .x in` Band `x notin A}`
`2. A Delta B = B Delta A` (commutative)

For example,
Let `A= {1, 2, 3, 4, 5}` and `B= {1, 3, 5, 7},` then `A-B= {2, 4}, B- A= {7}`
`therefore A Delta B = (A -B) uu (B- A) = {2, 4, 7}`