CmSc 175 Discrete Mathematics
SETS: Formal Definitions
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1. Relations between sets
1.1. Equality
A = B if and only if ∀x , x є A ↔ x є B
1.2. Subsets
A ⊆ B if and only if ∀x , x єA Æ x є B
A ⊂ B iff ∀x , x є A Æ x є B Λ ∃ x , x є B Λ x ∉ A
properties:
1.3. Disjoint sets
If A ⊆ B and B ⊆ A then A = B
∀A , Ø ⊆ A
∀A , A ⊆ U
A and B are disjoint if and only if ~∃ x, (x є A ) Λ (x є B)
i.e. ∀ x, x ∉ A V x ∉ B
A, B disjoint iff A ∩ B = Ø
2. Operations on sets
2.1. Intersection
A ∩ B = {x | (x є A ) Λ (x є B)}
properties:
A ∩ B ⊆ A,
A ∩Ø =Ø
A ∩ U =A
A∩ B⊆ B
A ∪ B = {x | (x є A) V (x є B)}
2. 2. Union
properties:
2.3. Difference
A∪B
A – B = { x | ( x є A) Λ ( x ∉ B)}
properties:
2.4. Complement
A⊆ A∪ B B ⊆
A ∪ Ø = A
A ∪ U =U
A–Ø =A
A–U =Ø
.
~A = {x | x ∉ A }
properties:
A ∪ ~A = U
A ∩ ~A = Ø
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