QUES: 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?

ANS: applying a to the first equality yields the second

QUES: Many teachers at Sabha were Egyptian, and for the first time Gaddafi had access to pan-Arab newspapers and radio broadcasts, most notably the Cairo-based Voice of the Arabs. Growing up, Gaddafi witnessed significant events rock the Arab world, including the 1948 Arab–Israeli War, the Egyptian Revolution of 1952, the Suez Crisis of 1956, and the short-lived existence of the United Arab Republic between 1958 and 1961. Gaddafi admired the political changes implemented in the Arab Republic of Egypt under his hero, President Gamal Abdel Nasser. Nasser argued for Arab nationalism; the rejection of Western colonialism, neo-colonialism, and Zionism; and a transition from capitalism to socialism. Nasser's book, Philosophy of the Revolution, was a key influence on Gaddafi; outlining how to initiate a coup, it has been described as "the inspiration and blueprint of [Gaddafi's] revolution."
What notable event occurred in 1952?

ANS: Egyptian Revolution

QUES: Together with caesium and gold (both yellow), and osmium (bluish), copper is one of only four elemental metals with a natural color other than gray or silver. Pure copper is orange-red and acquires a reddish tarnish when exposed to air. The characteristic color of copper results from the electronic transitions between the filled 3d and half-empty 4s atomic shells – the energy difference between these shells is such that it corresponds to orange light. The same mechanism accounts for the yellow color of gold and caesium.
How many metals have a natural color that isn't gray?

ANS: four

QUES: While there is some dispute as to the matter of influence, Popper had a long-standing and close friendship with economist Friedrich Hayek, who was also brought to the London School of Economics from Vienna. Each found support and similarities in the other's work, citing each other often, though not without qualification. In a letter to Hayek in 1944, Popper stated, "I think I have learnt more from you than from any other living thinker, except perhaps Alfred Tarski." Popper dedicated his Conjectures and Refutations to Hayek. For his part, Hayek dedicated a collection of papers, Studies in Philosophy, Politics, and Economics, to Popper, and in 1982 said, "...ever since his Logik der Forschung first came out in 1934, I have been a complete adherent to his general theory of methodology."
Who did Alfred Tarski say he had learned more from than any other living thinker?

ANS:
unanswerable