what is set and oprations

1. Set Theory
By : Muhammad Jibran
17-BS(IT)-16
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2. Basic Definitions
• A Set is a collection of well define items,
called set .
• Example:
• {1, 2, 3}
• S = {Ali, Rahman, Rashid}
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3. The Integers Set
• We define the Integers to be:
Z = {…, -2, -1, 0, 1, 2, 3, …}
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4. Natural Numbers
• Set of Natural Numbers
• N = {1,2,3,4,5,6,7,8,9,……}
• It is the sub set of the Whole Number set.
• Whole Number set is given as
• W={0,1,2.3.4.5.6.7.8……}
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5. Subsets
• If A and B are sets, A is called a subset of
B, denoted A ⊆ B, provided every element
of A is an element of B.
• So, A ⊆ B means ∀x, if x ∈ A, then x ∈ B.
• We also say, “A is contained in B” or
“B contains A” to show this relationship.
• Equivalently, we denote A ⊄ B provided
∃x ∋ x ∈ A and x ∉ B.
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6. Examples of Subsets
• If A = {1, 2, 3} and B = {0, 1, 2, 3, 4}, then
clearly A ⊆ B.
• {{1}, {2}} ⊆ {{1}, {2}, {1,2}}.
• Q ⊆ R and Z ⊆ Q and N ⊆ Z.
• {a, b, c} is a proper subset of {a, b, c, d}.
• {a, b, c} is an improper subset of {a, b, c}.
• We denote interval subsets of R as
[a, b) = {x ∈ R | a ≤ x < b}. So [2, 5) ⊆ [0,5].
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7. Set Equality
• We say sets A and B are equal (A = B) if
every element of A is in B and every element
of B is in A.
• Thus, A = B means A ⊆ B and B ⊆ A.
• For example {1, 2, 3} = {1, 2, 3}, but
A = {1, 2, 3} ≠ {1, 2, 3, 4} = B, since 4 ∈ B
but 4 ∉ A
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8. Operations on Sets
• Given sets A and B, which are subsets of a
universal set, U, we define the following:
• (Union) A ∪ B = {x ∈ U | x ∈ A or x ∈ B}.
• (Intersection) A ∩ B = {x ∈ U | x ∈ A and x ∈ B}.
• (Difference)
A − B = {x ∈ U | x ∈ A and x ∉ B}.
• (Complement) A’
= {x ∈ U | x ∉ A}.
• Note that A’
= U − A.
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9. Examples of Set Operations
• Let U = R, A = [1, 3] and B = (2, 4).
• A ∪ B = [1, 4)
• A ∩ B = (2, 3]
• A − B = [1, 2]
• B − A = (3, 4)
• Ac
= (−∞, 1) ∪ (3, ∞)
• Bc
= (−∞, 2] ∪ [4, ∞)
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10. The End
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