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 death duties in the 1940s, and the nationalization of their coal mines, greatly reduced the wealth of the Fitzwilliams, and most of the contents of the house were dispersed, in auction sales in 1948, 1986 and 1998, though many remain in the family, with many works loaned to museums by the " Trustees of the Fitzwilliam Estates ".

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 minutes later it came again -- a double explosion, followed by a third, sounding more distant.

The double editorial on Two Aspects Of `` The U.S. Spirit '' was subtly calculated to suggest a moral sanction for gambles great as well as small, reflecting popular approval of this questionable attitude toward the highest office in the land.

Two batters later, and to the horror of the packed stadium, Cubs shortstop Alex Gonzalez misplayed an inning ending double play, loading the bases and leading to eight Florida runs and a Marlin victory.

A second single, the double A-sided single " Two Doors Down "/" It's All Wrong, But It's All Right " also topped the country singles chart and crossed over to the pop top twenty.

Two significant recent works written for solo bass include, Mario Davidovsky's Synchronisms No. 11 for double bass and electronic sounds and Elliott Carter's Figment III, for solo double bass.

# On 15 September 1790, to his double first cousin Maria Teresa of the Two Sicilies ( 6 June 1772 – 13 April 1807 ), daughter of King Ferdinand I of the Two Sicilies ( both were grandchildren of Empress Maria Theresa and shared all of their other grandparents in common ), with whom he had twelve children, of whom only seven reached adulthood.

Two frequency converters would be used, and the radio would be a double conversion superheterodyne ; a common example is a television receiver where the audio information is obtained from a second stage of intermediate-frequency conversion.

Two revolutions ( two and a half for the axel ) is a " double " jump.

Two versions of transformer hybrids were used, the single transformer version providing unbalanced outputs with one end grounded, and the double transformer version providing balanced ports.

Two variants on the couched lance charge developed, the French method, en haie, with lancers in a double line and the German method, with lancers drawn up in a deeper formation which was often wedge-shaped.

Thus, the pharaohs were known as the rulers of the Two Kingdoms ( alternatively: Two Lands ), and wore the pschent, a double crown, each half representing sovereignty of one of the kingdoms.

In 1981, John Barton followed the 1957 Old Vic model and directed a heavily edited version of the play as a double bill with The Two Gentlemen of Verona for the RSC, starring Patrick Stewart as Titus, Sheila Hancock as Tamora, Hugh Quarshie as Aaron and Leonie Mellinger as Lavinia.

Bound double format with Between Two Worlds by Terry Carr

The golden age began with the 4th dynasty, which seems to have furthered the primary role of Memphis as a royal residence where rulers received the double crown, the divine manifestation of the unification of the Two Lands.

In Naples on 15 August 1790 by proxy and in Vienna on 19 September 1790 in person, Ferdinand married firstly his double first cousin, the Princess ( later Grand Duchess ) Luisa of Naples and Sicily ( 1773-1802 ), daughter of Ferdinand I of the Two Sicilies and Marie Caroline of Austria.

There have been other timpani concertos, notably, Philip Glass, considered one of the most influential composers of the late 20th century, wrote a double concerto titled Concerto Fantasy for Two Timpanists and Orchestra, which features its soloists playing nine drums apiece.

Their third single was a double A-side and featured the ballad, co-written by Cathy Dennis, " Two in a Million " and retro-styled, up-tempo " You're My Number One ".

Two rival explanations are put forward for the " double attestation " of Matthew and Luke regarding the virgin birth of Jesus:

The band documented this era in 1978 with Two for the Show, a double live album of recordings from various performances from its 1977 and 1978 tours.

* Splodgenessabounds reached No. 26 with the double A-side ' Two Little Boys ' / ' Horse ' in 1980.

For Malaco Records, Francis recorded a double album entitled " Where the Hits Are ", containing re-recordings of eighteen of her biggest hits as well as six classics of yesteryear Francis had always wanted to record, such as " Are You Lonesome Tonight ", or " Torn Between Two Lovers ".

Two years later, the duo became " true superstars " ( AllMusic ) with the release of their double CD Northern Exposure on mega-label Ministry of Sound.