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 ).

In Gregory ’ s day, history was not recognized as an independent field of study ; it was a branch of grammar or rhetoric, and historia ( defined as ‘ story ’) summed up the approach of the learned when they wrote what was, at that time, considered ‘ history .’ Gregory ’ s Dialogues Book Two, then, an authentic medieval hagiography cast as a conversation between the Pope and his deacon Peter, is designed to teach spiritual lessons.

Two independent sources attribute Dobrawa's important role in the conversion to Christianity of Mieszko I and Poland.

Two Cities Films, an independent production company also made some important films including This Happy Breed ( 1944 ), Blithe Spirit ( 1945 ) and Sir Laurence Olivier's Henry V ( 1944 ) and Hamlet ( 1948 ).

Two other groups, one at the University of Florida and another at the University of Pennsylvania, conducted independent clinical trials with similar results.

Two independent pathways for corrosion of elements are hydrolysis and oxidation by oxygen.

* Two Max Fleischer features, Gulliver's Travels ( 1939 ) and Hoppity Goes to Town ( 1941 ) were previously sold to NTA along with a few independent productions released by Paramount like One Third of a Nation ( 1939 ), as well as some features acquired by the studio after their theatrical release, including The Bells of St. Mary's ( 1945 ) and Good Sam ( 1947 ) ( both originally distributed by RKO Radio Pictures ).

Muhammad Ali Jinnah espoused the Two Nation Theory and led the League to adopt the Lahore Resolution of 1940, demanding the formation of independent states in the East and the West of British India.

Two independent evolutionary analyses of the peroxisomal proteome found homologies between the peroxisomal import machinery and the ERAD pathway in the endoplasmic reticulum, along with a number of metabolic enzymes that were likely recruited from the mitochondria.

: Two events A and B are independent if and only if

Two random variables X and Y are independent if and only if for every a and b, the events

Two redundant systems with independent failure modes, each having an MTBF of 100, 000 hours, could achieve a failure rate on the order of 10 < sup >− 10 </ sup > per hour because of the multiplication rule for independent events.

Two investigations, one a post-war official investigation, and the other an independent investigation by the Canadian Broadcasting Corporation, concluded that there was no evidence to support the story.

However, there are episodes that seem to contradict this view, or at least to point to a relaxation of it: " Genesis of the Daleks ," where the Time Lords ask the Doctor to prevent the creation of the Daleks or at least change them into a less violent race, " Image of the Fendahl ," where the Time Lords destroyed the fifth planet of Sol and then used a Time Loop to hide all records of its existence, and " Two Doctors ," where the Time Lords enlist the aid of the Doctor to prevent the independent development of their method of time travel.

Two of these leagues were associated with Organized Baseball ; the third and most recent incarnation was an independent league that operated for two seasons in 2002-03.

His first major film performance was in the independent film Where the Red Fern Grows ( Part Two ).

Two railroad lines intersect on the northwest side of town ; the question of whether the state of Minnesota could order installation of an interchange track between the two independent railroads led to a 1900 U. S. Supreme Court decision which affirmed that the state could indeed order the establishment of the " Hanley Falls Wye " for the public convenience, despite the railroads ' opposition.

Two television stations are based the Easton area: PBS affiliate WLVT Channel 39 in Bethlehem, and independent station WFMZ Channel 69 in Allentown.

Two non-profit, independent student newspapers publish at the University.

Two independent variants of the APOL1 gene found in African haplotypes carrying signatures of natural selection have been shown to confer protection against the acute version of sleeping sickness caused by T. b. rhodesiense, while at the same time increasing risk of kidney disease when inherited from both parents.

Two duos entered: Ina Brouwer ( former CPN ) combined with Mohammed Rabbae ( independent ), while Paul Rosenmöller ( independent ) formed a combination with Leoni Sipkes ( former PSP ); there were also five individual candidates, including Wim de Boer ( former chair of the PPR and member of the Senate ), Herman Meijer ( former CPN, future chair of the party ) and Ineke van Gent ( former PSP and future MP )).

; 1998: Two independent teams of NSF-supported astronomers discover that the expansion of the universe is actually speeding up, as if some previously unknown force, now known as dark energy, is driving the galaxies apart at an ever increasing rate.

* local or regional independent stations such as Citytv, CTV Two and CH,

Two sons, Domenico and Lorenzo, painted with him as his assistants and achieved some independent recognition.