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30 January, 18:29

Two sets are equal if they contain the

same elements. I. e., sets A and B are equal if

∀x[x ∈ A ↔ x ∈ B].

Notation: A = B.

Recall: Sets are unordered and we do not distinguish

between repeated elements. So:

{1, 1, 1} = {1}, and {a, b, c} = {b, a, c}.

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  1. 30 January, 18:50
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    Definition: Two sets are equal if they contain the

    same elements. I. e., sets A and B are equal if

    ∀x[x ∈ A ↔ x ∈ B].

    Notation: A = B.

    Recall: Sets are unordered and we do not distinguish

    between repeated elements. So:

    {1, 1, 1} = {1}, and {a, b, c} = {b, a, c}.

    Definition: A set A is a subset of set B, denoted

    A ⊆ B, if every element x of A is also an element of B.

    That is, A ⊆ B if ∀x (x ∈ A → x ∈ B).

    Example: Z ⊆ R.

    {1, 2} ⊆ {1, 2, 3, 4}

    Notation: If set A is not a subset of B, we write A 6⊆ B.

    Example: {1, 2} 6⊆ {1, 3}
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