³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980402761³2ip³2129.97.182.64³2data³2newauthor³31³3minor³30³3text³3A boolean algebra is a lattice (see LatticEs) which satisfies the following properties:

    There exists some element 0, such that av0=a for all a                 (bounded below)
    There exists some element 1, such that a^1=a for all a                 (bounded above)
    For all a,b,c, (avb)^c=(a^c)v(b^c)                                     (distributive law)
    For all a, there exists an element ~a such that av~a=1 and a^~a=0      (existence of complements)

Complements are guaranteed to be unique within bounded distributive lattices.  Note the definition of Boolean algebrae is very similar to that of RinGs, except elements have complements instead of inverses.  Moreover the distributive law can be shown to hold both ways, i.e. (a^b)vc=(avc)^(bvc).

A particularly important boolean algebra is given by the lattice {0,1} with ordering =>, defined so that 0=>1.  This is because this represents ordinary logical truth values: 0 is "false", 1 is "true", avb is "a or b", a^b is "a and b", and a=>b is "a implies b".

As an immediate consequence of the above, the MathematicalRelation''s'' over a set form a boolean algebra under these operations.  So, then, do the subsets of any given set under inclusions, and in fact the subobjects of ''any'' given mathematical object typically form a boolean algebra under inclusions.
³3summary³3*³2tscreate³2980402761³2keepts³2980403391³2id³21076³2host³2cnts2p64.uwaterloo.ca³2revision³21³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980403391³2ip³2129.97.182.64³2data³2text³3A boolean algebra is a lattice (see LatticEs) which satisfies the following properties:

    There exists some element 0, such that av0=a for all a                 (bounded below)
    There exists some element 1, such that a^1=a for all a                 (bounded above)
    For all a,b,c, (avb)^c=(a^c)v(b^c)                                     (distributive law)
    For all a, there exists an element ~a such that av~a=1 and a^~a=0      (existence of complements)

Complements are guaranteed to be unique within bounded distributive lattices.  Note the definition of Boolean algebrae is very similar to that of RinGs, except elements have complements instead of inverses.  Moreover the distributive law can be shown to hold both ways, i.e. (a^b)vc=(avc)^(bvc).

True-false statements form a boolean algebra, with implications => as the partial order.  Then avb represents "a or b",  a^b represents "a and b", and ~a represents "not a".  MathematicalRelation''s'' are a good example of such statements.
³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980402761³2keepts³2980407948³2id³21076³2host³2cnts2p64.uwaterloo.ca³2revision³22³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980407948³2ip³2129.97.182.33³2data³2newauthor³30³3minor³30³3text³3A boolean algebra is a lattice (see LatticEs) which satisfies the following properties:

    There exists some element 0, such that av0=a for all a                 (bounded below)
    There exists some element 1, such that a^1=a for all a                 (bounded above)
    For all a,b,c, (avb)^c=(a^c)v(b^c)                                     (distributive law)
    For all a, there exists an e
³3summary³3³2tscreate³2980402761³2keepts³2980407949³2id³21076³2host³2cnts2p33.uwaterloo.ca³2revision³23³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980407949³2ip³2129.97.182.33³2data³2text³3A boolean algebra is a lattice (see LatticEs) which satisfies the following properties:

    There exists some element 0, such that av0=a for all a                 (bounded below)
    There exists some element 1, such that a^1=a for all a                 (bounded above)
    For all a,b,c, (avb)^c=(a^c)v(b^c)                                     (distributive law)
    For all a, there exists an element ~a such that av~a=1 and a^~a=0      (existence of complements)

Complements are guaranteed to be unique within bounded distributive lattices.  Note the definition of Boolean algebrae is very similar to that of RinGs, except elements have complements instead of inverses.  Moreover the distributive law can be shown to hold both ways, i.e. (a^b)vc=(avc)^(bvc).
³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980402761³2keepts³2980408326³2id³21076³2host³2cnts2p33.uwaterloo.ca³2revision³24³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980408326³2ip³2129.97.182.33³2data³2newauthor³30³3minor³31³3text³3A boolean algebra is a lattice (see LatticEs) which satisfies the following properties:

    There exists some element 0, such that av0=a for all a                 (bounded below)
    There exists some element 1, such that a^1=a for all a                 (bounded above)
    For all a,b,c, (avb)^c=(a^c)v(b^c)                                     (distributive law)
    For all a, there exists an element ~a such that av~a=1 and a^~a=0      (existence of complements)

Complements are guaranteed to be unique within bounded distributive lattices.  Note the definition of Boolean algebrae is very similar to that of RinGs, except elements have complements instead of inverses.  Moreover the distributive law can be shown to hold both ways, i.e. (a^b)vc=(avc)^(bvc).

The PowerSet of any given set S forms a boolean algebra under the partial ordering "is a subset of", where 0={} and 1=S.
³3summary³3*³2tscreate³2980402761³2keepts³2980497082³2id³21076³2host³2cnts2p33.uwaterloo.ca³2revision³25³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980497082³2ip³2129.97.130.42³2data³2text³3A boolean algebra is a LatticE which satisfies the following properties:

    There exists some element 0, such that av0=a for all a                 (bounded below)
    There exists some element 1, such that a^1=a for all a                 (bounded above)
    For all a,b,c, (avb)^c=(a^c)v(b^c)                                     (distributive law)
    For all a, there exists an element ~a such that av~a=1 and a^~a=0      (existence of complements)

Complements are guaranteed to be unique within bounded distributive lattices.  Note the definition of Boolean algebrae is very similar to that of RinGs, except elements have complements instead of inverses.  Moreover the distributive law can be shown to hold both ways, i.e. (a^b)vc=(avc)^(bvc).

The PowerSet of any given set S forms a boolean algebra under the partial ordering "is a subset of", where 0={} and 1=S.
³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980402761³2keepts³2980497422³2id³21076³2host³2cnts3p42.uwaterloo.ca³2revision³26³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980497422³2ip³2129.97.130.42³2data³2newauthor³30³3minor³31³3text³3A boolean algebra is a LatticE which satisfies the following properties:

    There exists some element 0, such that av0=a for all a                 (bounded below)
    There exists some element 1, such that a^1=a for all a                 (bounded above)
    For all a,b,c, (avb)^c=(a^c)v(b^c)                                     (distributive law)
    For all a, there exists an element ~a such that av~a=1 and a^~a=0      (existence of complements)

Complements are guaranteed to be unique within bounded distributive lattices.  Note the definition of Boolean algebrae is very similar to that of RinGs, except elements have complements instead of inverses.  Moreover the distributive law can be shown to hold both ways, i.e. (a^b)vc=(avc)^(bvc).

The PowerSet of any given set S forms a boolean algebra under the partial ordering "is a subset of", where 0={} and 1=S.  Any subalgebra of this is called an algebra of sets.  If it turns out to be closed under arbitrary unions of sets, as opposed to just finite ones, it is called a TopOlogy.
³3summary³3*³2tscreate³2980402761³2keepts³2980497464³2id³21076³2host³2cnts3p42.uwaterloo.ca³2revision³27³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980497464³2ip³2129.97.130.42³2data³2text³3A boolean algebra is a LatticE which satisfies the following properties:

    There exists some element 0, such that av0=a for all a                 (bounded below)
    There exists some element 1, such that a^1=a for all a                 (bounded above)
    For all a,b,c, (avb)^c=(a^c)v(b^c)                                     (distributive law)
    For all a, there exists an element ~a such that av~a=1 and a^~a=0      (existence of complements)

Complements are guaranteed to be unique within bounded distributive lattices.  Note the definition of Boolean algebrae is very similar to that of RinGs, except elements have complements instead of inverses.  Moreover the distributive law can be shown to hold both ways, i.e. (a^b)vc=(avc)^(bvc).

The PowerSet of any given set S forms a boolean algebra under the partial ordering "is a subset of", where 0={} and 1=S.  Any subalgebra of this is called an algebra of sets.
³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980402761³2keepts³2980640398³2id³21076³2host³2cnts3p42.uwaterloo.ca³2revision³28³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980640398³2ip³2129.97.182.2³2data³2newauthor³30³3minor³30³3text³3A boolean algebra is a LatticE which satisfies the following properties:

    There exists some element 0, such that av0=a for all a                 (bounded below)
    There exists some element 1, such that a^1=a for all a                 (bounded above)
    For all a,b,c, (avb)^c=(a^c)v(b^c)                                     (distributive law)
    For all a, there exists an element ~a such that av~a=1 and a^~a=0      (existence of complements)

Complements are guaranteed to be unique within bounded distributive lattices.  Note the definition of Boolean algebrae is very similar to that of RinGs, except elements have complements instead of inverses.  Moreover the distributive law can be shown to hold both ways, i.e. (a^b)vc=(avc)^(bvc).

The PowerSet of any given set S forms a boolean algebra under the partial ordering "is a subset of", where 0={} and 1=S.  Any subalgebra of this is called an algebra of sets, and in particular, an algebra of sets closed under countable unions is called a TopOlogy.
³3summary³3*³2tscreate³2980402761³2keepts³2980640423³2id³21076³2host³2cnts2p02.uwaterloo.ca³2revision³29³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980640423³2ip³2129.97.182.2³2data³2text³3A boolean algebra is a LatticE which satisfies the following properties:

    There exists some element 0, such that av0=a for all a                 (bounded below)
    There exists some element 1, such that a^1=a for all a                 (bounded above)
    For all a,b,c, (avb)^c=(a^c)v(b^c)                                     (distributive law)
    For all a, there exists an element ~a such that av~a=1 and a^~a=0      (existence of complements)

Complements are guaranteed to be unique within bounded distributive lattices.  Note the definition of Boolean algebrae is very similar to that of RinGs, except elements have complements instead of inverses.  Moreover the distributive law can be shown to hold both ways, i.e. (a^b)vc=(avc)^(bvc).

The PowerSet of any given set S forms a boolean algebra under the partial ordering "is a subset of", where 0={} and 1=S.  Any subalgebra of this is called an algebra of sets, and in particular, an algebra of sets closed under arbitrary unions is called a TopOlogy.
³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980402761³2keepts³2980730109³2id³21076³2host³2cnts2p02.uwaterloo.ca³2revision³210