In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained in B. That is, all elements of A are elements of B. A and B may be equal; the relationship of one set being a subset of another is called inclusion or sometimes containment. A is a subset of B may be expressed as B includes A, or A is included in B; the subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, the subset relation itself is the Boolean inclusion relation. If A and B are sets and every element of A is an element of B A is a subset of B, denoted by A ⊆ B, or equivalently B is a superset of A, denoted by B ⊇ A. If A is a subset of B, but A is not equal to B A is a proper subset of B, denoted by A ⊊ B, or equivalently B is a proper superset of A, denoted by B ⊋ A. For any set S, the inclusion relation ⊆ is a partial order on the set P of all subsets of S defined by A ≤ B ⟺ A ⊆ B.

We may partially order P by reverse set inclusion by defining A ≤ B ⟺ B ⊆ A. When quantified, A ⊆ B is represented as ∀x. A set A is a subset of B if and only if their intersection is equal to A. Formally: A ⊆ B ⇔ A ∩ B = A. A set A is a subset of B if and only if their union is equal to B. Formally: A ⊆ B ⇔ A ∪ B = B. A finite set A is a subset of B if and only if the cardinality of their intersection is equal to the cardinality of A. Formally: A ⊆ B ⇔ | A ∩ B | = | A |; some authors use the symbols ⊃ to indicate subset and superset respectively. For example, for these authors, it is true of every set A that A ⊂ A. Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and proper superset respectively; this usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y x may or may not equal y, but if x < y x does not equal y, is less than y. Using the convention that ⊂ is proper subset, if A ⊆ B A may or may not equal B, but if A ⊂ B A does not equal B; the set A = is a proper subset of B =, thus both expressions A ⊆ B and A ⊊ B are true.

The set D = is a subset of E =, thus D ⊆ E is true, D ⊊ E is not true. Any set is a subset of itself, but not a proper subset; the empty set, denoted by ∅, is a subset of any given set X. It is always a proper subset of any set except itself; the set is a proper subset of The set of natural numbers is a proper subset of the set of rational numbers. These are two examples in which both the subset and the whole set are infinite, the subset has the same cardinality as the whole; the set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinite but the latter set has a larger cardinality than the former set. Another example in an Euler diagram: Inclusion is the canonical partial order in the sense that every ordered set is isomorphic to some collection of sets ordered by inclusion; the ordinal numbers are a simple example—if each ordinal n is identified with the set of all ordinals less than or equal to n a ≤ b if and only if ⊆. For the power set P of a set S, the inclusion partial order is the Cartesian product of k = |S| copies of the partial order on for which 0 < 1.

This can be illustrated by enumerating S = and associating with each subset T ⊆ S the k-tuple from k of which the ith coordinate is 1 if and only if si is a member of T. Containment order Jech, Thomas. Set Theory. Springer-Verlag. ISBN 3-540-44085-2. Media related to Subsets at Wikimedia Commons Weisstein, Eric W. "Subset". MathWorld

Sonoran Science Academy-Tucson is a public school, managed by Sonoran Schools, a charter management organization. The school is located in Tucson; the school is focused on STEM and college preparation. The school serves 800 students in grades K-12; the school colors are gold. The mascot is the Golden Eagle; the school is accredited by AdvancED. In August 2013, the Arizona Department of Education released A-F letter grades for all district and charter schools in the state. Letter grades are based on the weighting of student performance on the AIMS tests and student academic growth from year to year, along with additional points awarded for high English Language Learner reclassifications, significant reductions in dropout rates; every school and district received a report card with a grade that reflects their annual academic profile. In 2016, ADE awarded SSA a grade of "A." Sonoran Science Academy Tucson is operated by the Daisy Education Corporation which began in 1999. The founding board members established the Daisy Education Corporation because of a shared belief that excelling in math and science would prepare America's youth to achieve success in college, the workplace, the 21st century.

The founding members recognized that students in the United States were lagging behind their peers from other countries and that schools offering a rigorous, skill-level curriculum that focused on science, technology and math would provide the foundations to graduate independent thinkers who are capable of innovation and prepared to take leadership roles. Sonoran Science Academy Tucson called Daisy Science Academy, was opened in 2001, serving grades six through eight; the name of the school was changed the following year to Sonoran Science Academy and the school began adding grade levels each year until kindergarten through 12th grade were being served. Sonoran Science Academy participates in regional and national competitions, such as Intel ISEF, USAMO, FIRST Robotics Competition, National History Day, Academic Decathlon, MATHCOUNTS and Voice of Democracy. Sonoran Science Academy has been noted for the extracurricular achievement of its students. In 2012 Stan Palasek became the first Arizona student since 2007 to be awarded First Place at the Intel International Science and Engineering Fair for his work in molecular evolution.

In 2013 Augustus Woodrow-Tomizuka was recognized internationally with First Place at the Jacobs School of Music's international guitar competition. Students View Chivatanaporn from Sonoran Science Academy-Tucson and Nicholas Daniels from Sonoran Science Academy-Tucson received numerous National Honors from The Congressional Awards Program for achieving certificate and medal requirements from years 2014-2017. View and Nicholas were the first Sonoran Science Academy-Tucson students to achieve these national honors while attending the school. In 2016 and 2017, Pima County Supervisor Ally Miller today gave special recognition to the members of Sonoran Science Academy's FIRST Robotics team, CRUSH 1011, for winning the 2016 Colorado FIRST Robotics Team Award in Denver, Colorado and as 2017 FRC Houston World Champion. Supervisor Miller recognized CRUSH 1011 for their outstanding achievements by presenting them with a proclamation which declared the month of May as “STEM Education and CRUSH 1011 Robotics Team Appreciation Month.”First Robotic Competition Houston World Championship: In April 2017, CRUSH competed with 400 teams at the FRC Houston FIRST World Championship.

After a few rough matches on Thursday, CRUSH climbed in the ranks on Friday to end as the 8th seed. This was the first time. CRUSH was selected first overall to join FRC team 973 on the 1st seed alliance. Team 2928, 5499 joined our alliance. Our west coast alliance went undefeated in elimination matches, becoming Division Champions and advanced to Einstein Field. On Einstein, CRUSH had the opportunity to play against incredible teams, many of which have been consistent role models for CRUSH. In our 5 semi-final matches on Einstein, CRUSH won 3 matches and lost 2; because of the alliance's high scoring average, we made it into the Einstein Finals. After two tough matches, CRUSH and our alliance of 973, Greybots, 2928, Viking Robotics, 5499, The Bay Orangutans, were crowned the Houston World Champions. CRUSH was honored to bring home the Championship win to our state and school. CRUSH was the first FRC team from Arizona to win the Championship, one of the few who have made it to Einstein. Much of our success is due to the teams who have inspired and supported us over our 15 year history, none of it would have been possible without our sponsors and parents

Ana Maria Machado is a Brazilian writer of children's books, one of the most significant alongside Lygia Bojunga Nunes and Ruth Rocha. She received the international Hans Christian Andersen Medal in 2000 for her "lasting contribution to children's literature". Machado was born in Rio de Janeiro in 1941, she started her career as a painter in New York City. After studying Romance languages she did a PhD with Roland Barthes at the'École pratique des hautes études' in Paris, she worked as journalist for the BBC in London. In 1979, she opened the first children's bookshop in Brazil,'Malasartes'. In 1969, Ana Maria Machado started to write. "I belong to that generation of writers who began to write during the military dictatorship, as children’s literature, alongside poetry and song texts, were amongst the few literary forms with which, through the poetic and symbolic use of language, you could make the ideas of a joie de vivre, individual freedom and respect for human rights known." Her story Menina Bonita do laço de fita about a white and a black rabbit who marry and have a whole hoard of black and black and white patterned children, is a charming book about the living together of diverse ethnic groups.

In'Era uma vez um tirano' three children defy a tyrant who has forbidden colour and any happiness. Without pointing fingers, Ana Maria Machado always dresses up her messages in humorous stories and trusts the ability of her young readers to read between the lines. Similar to many Brazilian children's book authors of her generation, Ana Maria Machado is said to be in the tradition of the great children's book author, Jose Bento Monteiro Lobato, her writing is marked, in the style of "magical realism", by a subtle mix of social satire and fantastic elements as well as a conscious and playful use of language and narrative structures. In'História meio ao contrário', Ana Maria Machado turns the classic narrative structure of the fairy tale on its head and lets her story begin with: "And if they didn’t die they are still alive today" and end with "once upon a time". In'Bisa Bia, Bisa Bel', one of her central works, Isabell's internal dialogue with her dead great-grandmother, Bisa Bia, her own great-grandchild from the future, Bisa Bel, becomes a magical journey to the invisible connections between the generations, which allow Isabell to find her own way.

For the author, fantasy means to expand the sense for space and time and to allow reality and fantasy to mix with each other. Just as brilliantly in ‘Palavra de Honra’ Machado tells the story of a Luso-Brazilian family which has become wealthy since their arrival in the 19th century; the reader encounters Letícia, who tries to reconstruct her own story out of the dispersed remains of the family legacy. Ana Maria Machado lives with her family in Rio de Janeiro; the biennial Hans Christian Andersen Award conferred by the International Board on Books for Young People is the highest recognition available to a writer or illustrator of children's books. Jansson received the writing award in 2000. Ana Maria Machado has written more than hundred books for children and adults in 17 countries for which she has received the most significant Brazilian awards and many international honours. Alice e Ulisses, 1983 Tropical Sol da Liberdade, 1988 Canteiros de Saturno, 1991 Aos Quatro Ventos, 1993 O Mar Nunca Transborda, 1995 A Audácia dessa Mulher, 1999 Esta Força Estranha, 1998 Para Sempre, 2001 Official website University of San Francisco School of Education