Problem: Two groups G and H are called isomorphic if there exist group homomorphisms a: G H and b: H G, such that applying the two functions one after another in each of the two possible orders gives the identity functions of G and H. That is, a(b(h)) = h and b(a(g)) = g for any g in G and h in H. From an abstract point of view, isomorphic groups carry the same information. For example, proving that g g = 1G for some element g of G is equivalent to proving that a(g) a(g) = 1H, because applying a to the first equality yields the second, and applying b to the second gives back the first.
How can showing the second equality yields the first prove the concept of isomorphic groups?
The answer is the following: applying a to the first equality yields the second

Problem: In Arthurian legend, Avalon became associated with Glastonbury Tor when monks at Glastonbury Abbey claimed to have discovered the bones of King Arthur and his queen. What is more certain is that Glastonbury was an important religious centre by 700 and claims to be "the oldest above-ground Christian church in the World" situated "in the mystical land of Avalon." The claim is based on dating the founding of the community of monks at AD 63, the year of the legendary visit of Joseph of Arimathea, who was supposed to have brought the Holy Grail. During the Middle Ages there were also important religious sites at Woodspring Priory and Muchelney Abbey. The present Diocese of Bath and Wells covers Somerset with the exception of the Parish of Abbots Leigh with Leigh Woods in North Somerset and a small area of Dorset. The Episcopal seat of the Bishop of Bath and Wells is now in the Cathedral Church of Saint Andrew in the city of Wells, having previously been at Bath Abbey. Before the English Reformation, it was a Roman Catholic diocese; the county now falls within the Roman Catholic Diocese of Clifton. The Benedictine monastery Saint Gregory's Abbey, commonly known as Downside Abbey, is at Stratton-on-the-Fosse, and the ruins of the former Cistercian Cleeve Abbey are near the village of Washford.
The Episcopal seat of the Bishop is now where
The answer is the following: The Episcopal seat of the Bishop of Bath and Wells is now in the Cathedral Church of Saint Andrew in the city of Wells, having previously been at Bath Abbey

Problem: In 1989, the first full season with night baseball at Wrigley Field, Don Zimmer's Cubs were led by a core group of veterans in Ryne Sandberg, Rick Sutcliffe and Andre Dawson, who were boosted by a crop of youngsters such as Mark Grace, Shawon Dunston, Greg Maddux, Rookie of the Year Jerome Walton, and Rookie of the Year Runner-Up Dwight Smith. The Cubs won the NL East once again that season winning 93 games. This time the Cubs met the San Francisco Giants in the NLCS. After splitting the first two games at home, the Cubs headed to the Bay Area, where despite holding a lead at some point in each of the next three games, bullpen meltdowns and managerial blunders ultimately led to three straight losses. The Cubs couldn't overcome the efforts of Will Clark, whose home run off Maddux, just after a managerial visit to the mound, led Maddux to think Clark knew what pitch was coming. Afterward, Maddux would speak into his glove during any mound conversation, beginning what is a norm today. Mark Grace was 1117 in the series with 8 RBI. Eventually, the Giants lost to the "Bash Brothers" and the Oakland A's in the famous "Earthquake Series."
Who did the Cubs meet in the NLCS during 1989?
The answer is the following: San Francisco Giants

Problem: Executive Directors and Under-Secretaries General of the UN
2011present Dr Babatunde Osotimehin (Nigeria)
20002010 Ms Thoraya Ahmed Obaid (Saudi Arabia)
19872000 Dr Nafis Sadik (Pakistan)
196987 Mr Rafael M. Salas (Philippines)
Who was Executive Director or Under-Secretary General from 2000 to 2010?
The answer is the following:
Thoraya Ahmed Obaid