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 of the new hands, a Mexican named Jose Amado and a kid known only as Laredo, were picked for the first trick of riding night herd.

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 facets of this aspect of the literary process have special significance for our time.

Daniel Baker deluged his `` Unckle Quyne '' with requests to pay money for him to drapers in Watling Street and at the Two Cats in Canning Street.

Two days later Mr. Rayburn left Washington for the last time.

Two things have happened in recent months to bring the Council into perspective: each provides a basis for renewed hope and joy.

Two special issues were published, one for November 1959 on Space Medicine, the other the Tenth Anniversary issue for January 1960.

Two data centers have been established for the collection, indexing, critical evaluation, and dissemination of bibliographies and critical values in the fields of transition probabilities and collision cross sections.

Two committees of members of the Advisory Board constitute the committees of selection -- one for the selection of Fellows from Canada, the United States, and the English-speaking Caribbean area and one for the selection of Fellows from the Latin American republics and the Republic of the Philippines.

Two pyrometers shown in figure 1 and 2 ( Pyrometer Instrument Co. Model 95 ) served for simultaneous measurement of the anode surface temperature and the temperature distribution along the anode holder.

Two very useful ways for modifying a form-dictionary are the addition to the dictionary of complete paradigms rather than single forms and the application of a single change to more than one dictionary form.

Two metabolites ( 1, and 2 ) ) of p-aminobenzoic acid ( PABA ) which act as cofactors for the hydroxylation of aniline by acid-fast bacteria are biosynthesized from Aj.

Two martinis for lunch -- that was the trouble.

Two millions were added to what had been set aside for it in Mrs. Meeker's lifetime, and the proviso made that as long as Brian Thayer continued to discharge his duties as administrator of the fund to the satisfaction of the board of trustees ( hereinafter appointed by the bank administering the estate ) he was to be retained in his present capacity at a salary commensurate with the increased responsibilities enlargement of the fund would entail.

-- 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 members of the Democratic-endorsed majority on the school board said they probably would vote to appeal a ruling by the state Board of Education, which said yesterday that the school committee acted improperly in its appointment of the coordinator, Francis P. Nolan 3rd, the Democratic-endorsed committee chairman, could not be reached for comment.

The penalty for this is noticeable in the big, bold, brilliant, but brassy piano sounds in Melody And Percussion For Two Pianos ( SP-44007 ).

Two sharply contrasting places designed for public enjoyment are now on display.

Two letters had arrived for Miss Theresa Stubblefield: she put them in her bag.

* 1951 – Freddie Wadling, Swedish singer-songwriter, guitarist, and actor ( Leather Nun and Blue for Two )

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.

Writer James Russell Lowell referred to Alcott in his poem " Studies for Two Heads " as " an angel with clipped wings ".

Two locations on the Moon were given primary consideration for exploration by the Apollo 16 expedition, the Descartes Highlands region west of Mare Nectaris and the crater Alphonsus.