Basic relationships between sets

1.4

• Two sets S and T are said to be equal iff they have the same elements:

S=T iff ?x (x?S ? x?T)

Examples: {a,b}={b,a}

{a,b,b}={a,b}

• S is a subset of T iff every element of S is also an element of T:

S?T iff ?x (x?S ? x?T)

Examples: {a,b} ? {a,b,c} {a,b} ? {a,b}

Note that S=T iff S?T ? T?S

• S is a proper subset of T iff S is a subset of T but not vice versa:

S?T iff S?T ? T? S

or

S?T iff S?T ? S?T

• S and T are disjoint iff they have no common elements:

??x (x?S ? x ?T)

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