This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are tips for identifying set theory problems and the solutions to the questions: Tips to Identify Set Theory Problems Set theory problems deal with collections of objects, called sets, and the relationships and operations between them. Here are some common indicators: Symbols: Look for specific set notation symbols: (Union): "or", "at least one", "all elements in either set". (Intersection): "and", "both", "common elements". ' or ^c (Complement): "not in", "elements outside the set". or - (Difference): "in A but not in B". (Subset): "is contained in". (Element of): "is a member of". (Empty set): A set with no elements. U (Universal set): The set of all possible elements under consideration. Keywords: Phrases like "equivalence of sets," "simplify the set expression," "prove the identity," "number of elements," "survey results," "groups of people," "at least one," "neither," "only." Context: Problems involving grouping, categorization, or relationships between different collections of items (e.g., students taking different subjects, people with different hobbies). Venn Diagrams: If the problem mentions or can be easily visualized with overlapping circles, it's likely a set theory problem. Solutions to the Questions 4. The equivalence of A (B C) is: a) (A B') (A C) b) (A B) C c) (A B) (A C) d) (A B) (A C) This question asks for an equivalent expression using the Distributive Law of set operations. The Distributive Law states that for any sets A, B, and C: A (B C) = (A B) (A C) This law is analogous to distributing multiplication over addition in algebra, e.g., a × (b + c) = (a × b) + (a × c). Comparing the given expression with the options, option (c) directly matches the Distributive Law. The correct option is c) (A B) (A C). 5. For an AP, a=25, d=-3 find the value of n if S_n = 112. Step 1: Identify the given values for the Arithmetic Progression (AP). First term: a = 25 Common difference: d = -3 Sum of the first n terms: S_n = 112 Step 2: Use the formula for the sum of the first n terms of an AP. The formula is: S_n = (n)/(2) [2a + (n-1)d] Step 3: Substitute the given values into the formula. 112 = (n)/(2) [2(25) + (n-1)(-3)] Step 4: Simplify the equation. 112 = (n)/(2) [50 - 3(n-1)] 112 = (n)/(2) [50 - 3n + 3] 112 = (n)/(2) [53 - 3n] Multiply both sides by 2 to eliminate the fraction: 2 × 112 = n(53 - 3n) 224 = 53n - 3n^2 Step 5: Rearrange the equation into a standard quadratic form (Ax^2 + Bx + C = 0). 3n^2 - 53n + 224 = 0 Step 6: Solve the quadratic equation for n using the quadratic formula n = -B ± sqrt(B^2 - 4AC)2A. Here, A=3, B=-53, C=224. n = -(-53) ± sqrt((-53)^2 - 4(3)(224))2(3) n = 53 ± sqrt(2809 - 2688)6 n = 53 ± sqrt(121)6 n = (53 ± 11)/(6) This gives two possible values for n: n_1 = (53 + 11)/(6) = (64)/(6) = (32)/(3) n_2 = (53 - 11)/(6) = (42)/(6) = 7 Since the number of terms (n) must be a positive integer, n = (32)/(3) is not a valid solution. Therefore, the value of n is 7. The value of n is 7.