In mathematics, the associative property is a property of some binary operations. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed; that is, rearranging the parentheses in such an expression will not change its value. Consider the following equations: + 4 = 2 + = 9 2 × = × 4 = 24. Though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real numbers, it can be said that "addition and multiplication of real numbers are associative operations". Associativity is not the same as commutativity, which addresses whether or not the order of two operands changes the result. For example, the order does not matter in the multiplication of real numbers, that is, a × b = b × a, so we say that the multiplication of real numbers is a commutative operation.

Associative operations are abundant in mathematics. However, many important and interesting operations are non-associative. In contrast to the theoretical properties of real numbers, the addition of floating point numbers in computer science is not associative, the choice of how to associate an expression can have a significant effect on rounding error. Formally, a binary operation ∗ on a set S is called associative if it satisfies the associative law: ∗ z = x ∗ for all x, y, z in S. Here, ∗ is used to replace the symbol of the operation, which may be any symbol, the absence of symbol as for multiplication. Z = x = xyz for all x, y, z in S; the associative law can be expressed in functional notation thus: f = f. If a binary operation is associative, repeated application of the operation produces the same result regardless of how valid pairs of parentheses are inserted in the expression; this is called the generalized associative law. For instance, a product of four elements may be written, without changing the order of the factors, in five possible ways: d d a a If the product operation is associative, the generalized associative law says that all these formulas will yield the same result.

So unless the formula with omitted parentheses has a different meaning, the parentheses can be considered unnecessary and "the" product can be written unambiguously as a b c d. As the number of elements increases, the number of possible ways to insert parentheses grows but they remain unnecessary for disambiguation. An example where this does not work is the logical biconditional ↔, it is associative, thus A ↔ is equivalent to ↔ C, but A ↔ B ↔ C most means, not equivalent. Some examples of associative operations include the following; the concatenation of the three strings "hello", " ", "world" can be computed by concatenating the first two strings and appending the third string, or by joining the second and third string and concatenating the first string with the result. The two methods produce the same result. In arithmetic and multiplication of real numbers are associative.

The First Congregational Church known as Sioux City Baptist Church and most as Iglesia Pentecostes Evangelica Principe de Paz, is a house of worship located in Sioux City, United States. An architectural rarity, it is one of a small group of churches in the Prairie School style of architecture. Designed in the Prairie style with some eclectic touches by architect William L. Steele, its horizontal lines are emphasized by Roman brick and crisp rectilinear forms. Somewhat at variance are the distinctive dome and the prominent round heads on the windows. Fresh from his triumph with the Woodbury County Courthouse in collaboration with George Grant Elmslie, drawing on lessons learned during that collaboration, Steele built the church in 1916-1918; this church and the courthouse are the only two Prairie style buildings that are known to have a dome. It was built for a Congregational church, established in Sioux City back in 1857, replacing a more traditional church that had burned down in 1916. In 1966, that congregation built a new structure on Hamilton Boulevard.

First Congregational Church sold the building to Sioux City Baptist Church c. 1968. It was listed on the National Register of Historic Places in 1979 under that name. In 2009, Sioux City Baptist Church acquired a building on Viking Drive. Most the building has become the Iglesia Evangelica Pentecostes Principe de Paz, with services in Spanish aimed at the local Hispanic community. Due to the need for extensive building restoration and maintenance, the structure has been named to endangered building lists by at least two historic preservation groups. William L. Steele Prairie School Hartington City Hall and Auditorium Woodbury County Courthouse Unity Temple Stewart Memorial Presbyterian Church St. John African Methodist Episcopal Church

Iloilo City the City of Iloilo is a urbanized city on the southeastern tip of Panay island in the Philippines. It is the capital city of the province of Iloilo where it is geographically situated but, in terms of government and administration, it is politically independent. In addition, it is the center of the Iloilo-Guimaras Metropolitan Area, as well as the regional center and primate city of the Western Visayas region. In the 2015 census, Iloilo City had a population of 447,992 inhabitants, with a 1.02% population annual growth rate. For the metropolitan area, the total population is 946,146 inhabitants. Iloilo City is bordered by the towns of Oton in the west, Pavia in the north and Leganes in the northeast. Just across the Iloilo Strait in its eastern and southern coastlines, are the towns of Buenavista and Jordan in the island-province of Guimaras; the city was a conglomeration of former towns, which are now the geo-political districts consisting of: Villa Arevalo, Iloilo City Proper, Jaro, La Paz and Molo.

The district of Lapuz, a former part of La Paz, was declared a separate district in 2008. Iloilo City is the regional hub of education, tourism, culture and economy in Western Visayas. Famous food that came principally from Iloilo are Lapaz Batchoy, it is the Regional Center of National Government Offices, financial institutions and government-owned and controlled corporations. It is aiming for " Smart City of the Future " using technology of Internet of things for efficient delivery of services such as transportation safety, emergency response, environmental management etc. Parks and greenery are continue to be built bagging the ASEAN Clean Tourist City. Iloilo City is the Center of Economy where number of banks with most number of savings accounts third in the Philippines, 4th busiest airport and with on-going building construction frenzy. Iloilo city has the lowest crime rate in entire Philippines, lowest level of corruption, highest life expectancy in Visayas and Mindanao, huge concentration of middle class and rank 1 in happiness index, the most business-friendly city According to Spanish written accounts, the inhabitants of Panay island were hailed from Borneo and Sumatra.

Panay might be named after the kingdom of Pannai, located in Sumatra, since i and y are interchangeable in Spanish. Proof for this is corroborated by linguistic evidence such as Talang Tuo inscription where many Old Malayan words during Srivijayan survived in modern Ilongo Language; the Srivijayan City states of Sumatra was ruled by an Indian King Mahārāja with their Datu, a lesser vassal Malayan chieftain. Cruel rule of a certain Rajah Makatunao, the ten Bornean datus sailed to the island which they named Panay, Pani or Panae as part of Visayan Empire; the Malayan Datus offered a golden hat and a long pearl necklace called Manangyad in the Hiligaynon language as a barter to the Ati to allow them entry and habitation of the island. The Datus founded the Kadatuan Srivijaya consisted of several settlements called wanua/Benua/Banwa as corroborated by Kota Kapur inscription; some historians affirm the Sumatran origin of the people of Panay, observing that the Visayans derived their writing system from those of Toba, Celebes, Ancient Java and from the Edicts of the ancient Indian emperor Ashoka.

The probable proof of Sumatran origin of the Malay settlement in Panay is the account of P. Francisco Colin, S. J, a historian who came to Asia during the early years of the Spanish conquest of the Philippines; the following is his personal observation recorded during his visit to Sumatra: "In the middle of Sumatra, there is a spacious and extensive lake, around the shore of which many and several ethnic groups settle from where, in the past, there was a forced exodus of inhabitants to sail to and to settle in various islands. One of these ethnic groups was subjugated there and they were unable to flee for various circumstances. Someone speaking Pampango found out that they were not speaking Pampango among themselves, but they donned the old Pampango ethnic costume, and when he addressed an old man among them, the replied: You are descendants of the lost, that in times past left this place to settle in other lands, nothing was heard about them again." So, Colin concluded that the Tagalogs and Pampangos, other political or ethnic groups, by symbols used in expressing language, by color of dress and costume, one can believe that these came from parts of Borneo and Sumatra.

The phrase "subjugated ethnic groups" gives hint to the encroaching of the Majapahit Empire into the falling Srivijaya Empire, or to the Islamization of Sumatra, forcing the inhabitants to look for safer territories where they could preserve their freedom and culture. This resonates with the local Panay tradition regarding the arrival of ten Datus from Borneo; the kingdom of Pannai was a militant-nation allied under the Srivijaya Mandala that defended the conflict-ridden Strait of Malacca, the world's busiest maritime choke-point. The Visayan lore says that in the 13th century, ten Bornean datus came to the island which they named Panay, Pani or Panae. This, after they dissented from the unjust rule of a certain Rajah Makatunao and exiled themselves. Upon arriving on the island of Panay, they gave a golden hat and a long pearl necklace called Manangyad in the Hiligaynon language (meaning a long necklace that touches or "nagas