Two sets A and B are defined to be equal when they have precisely the same elements, that is, if every element of A is an element of B and every element of B is an element of A.

Two sets are said to have the same cardinality or cardinal number if there exists a bijection ( a one-to-one correspondence ) between them.

Two sets have the same cardinal number if and only if there is a bijection between them.

Two such functions are deemed equivalent when their respective sets of fixpoints have the same cardinality, corresponding to cycles of length one in a permutation.

* 1860 – Giuseppe Garibaldi sets sail from Genoa, leading the expedition of the Thousand to conquer the Kingdom of the Two Sicilies and giving birth to the Kingdom of Italy.

Two or three sound stages and four outdoor sets were destroyed ; fortunately, the rest of the Studios were still intact.

The centerpiece of the deregulation was the establishment of two categories of basic services licenses: Local loop ( LL ), for fixed line telecommunication within the 14 PTCL regions, and Long-distance and International ( LDI ), for connectivity between regions .” Two sets of criteria set by the regulatory authorities must be met before an operator is allowed to start operation: one for the issuance of a license and another for the maintenance of service quality.

Two blocks with a rope attached to one of the blocks and threaded through the two sets of pulleys form a block and tackle.

Two sets of rules exist, each of which can be played with a variety of layouts.

The Deep Cameo versions of proofs from 1965 to 1971 and 1981 Type Two are highly valued because of their scarcity, high grade examples of quarters from certain years of the 1980s ( such as 1981 – 1987 ) because of scarcity in high grades due to high circulation and in 1982 and 1983 no mint sets were produced making it harder to find mint state examples, and any coin from 1981 – 1994 graded in MS67 is worth upwards of $ 1000.

* Bruce Quarrie-wrote rule sets for Napoleonic and World War Two wargames and also on military history.

Two 1-connected paths can be consolidated together if the two groups of stones they start and end in are the same and their empty-point sets do not overlap.

Two sets of accents are spoken in the West Country, Cornish ( primarily in South Cornwall ) and West Country ( the counties of Devon, Somerset, Gloucestershire, Bristol, Dorset ( Not so much in east Dorset ), Wiltshire ( again, less so in eastern Wiltshire ), and also in East Cornwall ).

Two sets of 6 attitude control nitrogen jets were mounted on the ends of the solar panels.

Two disjoint sets.

Two sets are called similar if one is the image of the other under such a similarity.

Two other sets of units, Gaussian and CGS-EMU, are the same for magnetic properties and are commonly used in physics.

The third, " Coursing between the Two Poles ", involves three sets of movements which aim to enable the expulsion of bad energy ( e. g. pathogenic or black qi ) and the absorption of good energy into the body.

Two sets of extremists had arisen: Northerners who demanded no new slave territories under any circumstances, and Southerners who demanded free entry for slavery into all territories, the penalty for denial to be secession.

Two different sets of numbers bear this name: the Stirling numbers of the first kind and the Stirling numbers of the second kind.

Two forces converged to initiate the PDB: 1 ) a small but growing collection of sets of protein structure data determined by X-ray diffraction ; and 2 ) the newly available ( 1968 ) molecular graphics display, the Brookhaven RAster Display ( BRAD ), to visualize these protein structures in 3-D.

Two sets of brothers, Jacob and Daniel Beck and George and Daniel Miller first stopped in Williamsport before moving further to the north.

Battle Kings reappeared on the market in 2006, but not as King Size models, but rather as a name of military-oriented Two Pack-style sets of regular size models.

Two sets of pronunciations for the numerals exist in Japanese: one is based on Sino-Japanese ( on ' yomi ) readings of the Chinese characters and the other is based on the Japanese yamato kotoba ( native words, kun ' yomi readings ).

-- Two Americans and seven Cubans were executed by firing squads today as Castro military tribunals began decreeing the death penalty for captured invasion forces and suspected collaborators.

Two other reasons for this title are that he would support his aged father-in-law with his hand at Senate meetings, and that he had saved those men that Hadrian, during his period of ill-health, had condemned to death.

Two decades after Frank Herbert's death, his son Brian Herbert, along with Kevin J. Anderson, published two sequels – Hunters of Dune ( 2006 ) and Sandworms of Dune ( 2007 ) – based on notes left behind by Frank Herbert for what he referred to as Dune 7, his own planned seventh novel in the Dune series.

Two days passed between his death and the announcement of the election of Yuri Andropov as the new General Secretary, suggesting to many outsiders that a power struggle had occurred in the Kremlin.

Two inquisitions were taken after Oxford's death, the first in 1604 for his property in Essex, the second in 1608 for his Great Garden property in London.

Two days after her death, the lights of the Empire State Building were extinguished for 15 minutes in her memory.

Two thousand years after his death, Imhotep's status was raised to that of a deity of medicine and healing.

Two days after Tycho's unexpected death on October 24, 1601, Kepler was appointed his successor as imperial mathematician with the responsibility to complete his unfinished work.

A German Requiem was partially inspired by his mother's death in 1865 ( at which time he composed a funeral march that was to become the basis of Part Two, Denn alles Fleisch ), but it also incorporates material from a symphony which he started in 1854 but abandoned following Schumann's suicide attempt.

Two years after Brando's death, he " reprised " the role of Jor-El in the 2006 " loose sequel " Superman Returns, in which both used and unused archive footage of him as Jor-El from the first two Superman films was remastered for a scene in the Fortress of Solitude, and Brando's voice-overs were used throughout the film.

Two months after his death, the Poles offered to do homage to either Mary or Hedwig, on condition that the queen and her husband agree to live in Poland.

* 1950 – Two Canadian National Railway trains collide in northeastern British Columbia in the Canoe River train crash ; the death toll is 21, with 17 of them Canadian troops bound for Korea.

Two songs, " Out of Time " and " Beyond Hurt – Beyond Help ," were originally written by Darby Crash and Pat Smear prior to Crash's death, but were never recorded.

Two weeks after his death, Viola Liuzzo, a Unitarian Universalist civil rights activist, was murdered by white supremacists after her participation in the protest march from Selma to Montgomery, Alabama.

* 29 March: Two elderly men were stabbed to death while on their way to the synagogue for morning prayers, when a Palestinian gunman infiltrated the Neztarim settlement in the Gaza Strip.

Two months later, after the death of her mother, Queen Elizabeth The Queen Mother, Margaret's ashes were interred beside the bodies of her parents in the George VI Memorial Chapel at St George's Chapel, Windsor Castle.

Two centuries after his death, Hals received a number of ' posthumous ' students.

Two weeks later, however, his great excitement was crushed by his daughter Mair's death from appendicitis.

Two months later, on 21 October 1422, he became King of France upon his grandfather Charles VI's death in agreement with the Treaty of Troyes in 1420.

Two months after his death the Paris Academy made their award.

Two other important financial institutions, the Court of Wards and the Court of First Fruits and Tenths, owed their existence to him, although they were not set up until after his death.

Two days after Leonid Brezhnev's death, on 12 November 1982, Andropov was elected General Secretary of the CPSU, the first former head of the KGB to become General Secretary.

Two further trials, in 1450 and 1456, led to Jacopo's imprisonment on Crete and his eventual death there.

Two days after his father's death, the Somali National Alliance selected him to become the new president of the Republic of Somalia.

" In 1990 he teamed up with his co-panellist Barry Cryer in their own show Two old Farts in the Night, performing to full audiences at the Edinburgh festival, the Albert Hall, and the Festival Hall, touring the country irregularly until Rushton's death.