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2. ∅ ⊆ S, for every set S. S ⊆ S, for every set S. ༉ Be careful: A ⊆ B means “A (set) is a subset of B (set)” but a ∈ A means “a (member) belongs to A (set)”. A is or is not a Subset of B ༉ Showing that A is a Subset of B: to show that A ⊆ B, show that if x belongs to A, then x also belongs to B. ༉ Showing that A is not a Subset of B: to show that A ⊈ B, ﬁnd an element x ∈ A such that x ∉ B. Such an x is a counterexample to the claim that x ∈ A implies x ∈ B. Examples: 1. The set of all computer science majors at school “S” is a subset of all students at school “S”. 2. The set of all computer science majors at school “R” is not a subset of all students at school “S”. Another look at Equality of Sets ༉ Recall that two sets A and B are equal, denoted by A = B, if and only if ༉ Using logical equivalences we have that A = B if and only if ༉ This is equivalent to A ⊆ B and B ⊆ A Proper Subsets Deﬁnition: if A is a subset of B, but A ≠ B, then we say A is a proper subset of B, denoted by A ⊂ B. If A ⊂ B, then is true. B Venn Diagram A U Set Cardinality Deﬁnition: if there are exactly n distinct elements in S where n is a nonnegative integer, we say S is ﬁnite, otherwise, it is inﬁnite. Deﬁnition: the cardinality of a ﬁnite set A, denoted by A, is the number of (distinct) elements of A. Examples: 1. ø = 0 2. Let S be the letters of the English alphabet. Then S = 26 3. {1,2,3}...
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 Spring '14
 M.Nojoumian
 Computer Science, Natural number

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