Let A = {e, f, h}, then write all the permutations of A.

Q: Consider the following permutations in S,: a = (! 2 3 4 5 6 7 6 2 4 1 7 5 3) The order of B is (1 2…

A: 1)here o(beeta ) is 7

Q: 2. Let A Ø. If A has exactly n elements, then A has exactly 2" subsets, Vn € N.

A: A is non empty subset of n elements ,then A has exactly 2n subsets,∀n∈ℕ.

Q: Show that a permutation o E S, commutes with the transposition (a b) if and only we are in one of…

A: Given: f:S5→S5 is an automorphism. σ∈S5 To do: (d) Prove that σ commutes with the transposition ab…

Q: Let A and (A, n2 1) be subsets of 2 An - A as n → 00 =P(A,) P(A). as n 0o,

A: The given question is related with probability. Given that A and An , n≥1 are subsets of Ω . We have…

Q: Q1// Let Ai be even permutation in Sn. Then O(An)=n!/2

A: Even permutation:- element with even number of transportation belongs to even permutation set

Q: 48. Given {W, X, Y, Z}, a. List all the permutations of three elements from the set. b. List all the…

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Q: 8. Let A = {Al, Beth}and B = {up, down, left, right}. Fine %3D

A: # we are entitled to solve one question at a time, please resubmit the other questions if you wish…

Q: a b = a - b is a commutative binanry operation on Z

A: Commutative operation means without changing the operation the values doesn't change.

Q: Compute each of the following permutations. (a) (1423)(34)(56)(1324); (b) (1254)^2(123)(45); (c)…

A: Since you have asked a question with multiple subparts, we will solve the first three  subpart…

Q: Consider the following permutations in S,: 4 5 6 7 2 3 6 2 4 1 7 5 3/ The order of a is 4 OOO

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Q: Let a, = 1 and a₁ + a1 = n² + 2n, n ≥2. Evaluate S

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Q: Exercise 4. Consider the following permutations in S, _(1 2 3 4 5 6 7 8 9) (3 5 2 1 4 6 9 7 8) (1 2…

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Q: If S is a ssubset of T , then u* (T) sµ* (S). ulgn O İhi

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: abc and bca are the same permutation * T.

A: yes , ABC and BCA are the same permutation.

Q: Let a and ß belong to S,. Prove that a-'ß-laß is an even permutation. n°

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Q: 1. Let A, B and C be members of C. Compare An(BuC) and (A

A: Suppose there are two sets A and B then, the union (∪) of these two sets is defined as a set…

Q: d. a specific element of R° that is not in V. е. a specific element of v in V for which -v is not in…

A: Given (a,b,c) where c=a-2b

Q: Based on the choices below, is the combinations is a subset of the permutations. a. True b. False

A: To check: Whether the statement is true or false. Given: The statement is "the combinations is a…

Q: Prove this: A≤B iff AnB=A

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Q: 1) Find the number of permutations of all the letters of the w following conditions

A: “Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: Let f and g be permutations. Show that the order of h=fgf^−1 is the same as the order of g.

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Q: (e) There exist subsets A and B of N such that (A-B) n (ANB) = TRUE FALSE

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Q: 5(a) Let V = {[ :a, b eR}with standard m

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Q: Consider the following permutations in S: a-C ? 3 4 5 67 6 2 7 1 45 3 2 34 56 2 3 4 3 27 their…

A: here option(b) is true...none of these .

Q: Prove that if a|b and b|c, then a|c.

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Q: Prove that if n 2 3, the only element o of S, satisfying oy =ya for all y E S, is the identity…

A: Let n≥3. Let σ∈Sn such that σγ=γσ for all γ∈Sn. Suppose σ≠i, the identity Then there exists…

Q: Which of the following refers to the different arrangements of objects where some of them are the…

A: Permutation: Two different arrangements of objects where some of them are the same are identical are…

Q: Consider the following permutations in S,: (1 2 3 4 5 6 7 4 5 6 7 7 1 5 6) 1 3 a = 6 2 7 their…

A: To find the product of given permutations

Q: 10.

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Q: Suppose a, b, c, d e Z, a + b = c, and dla. Prove that d|b if and only if d|c. 5.

A: see below the calculation steps

Q: For all s, Isin s≤ 1

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Q: Let 1 2 3 4 5 6 7 8 2 35 6 7 1 8 4 :) 1 2 3 4 5 6 7 8 4 8 3 612 5 7 and be permutations on the set…

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Q: Let A = a22 Show that if d = a¡|a» – A2jɑ12 # 0, then 1 A- = a22 -a12

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Q: Let ? and ? be permutations on the set {1,2,3,4,5,6,78,9}, defined as follows

A: Introduction: A permutation is a bijective mapping from a set to itself. The general way to…

Q: Prove for all m, n, and k in w, that (m⋅n)k = mk.nk.

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Q: 1. Which of the following permutations in S, is odd? (a) 53124 (b) 51423 (c) 51234 (d) 54231

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Q: If two rows of A are interchanged to produce B, then det(B) = –det(A). \\ Big True fals e

A: We use properties of determinant

Q: Show that a permutation o E S; commutes with the transposition (a b) if and only we are in one of…

A: Hi! As per norms, we will be answering only question (d). If you need an answer to others then…

Q: Prove: If A ∪ C = B and A ∪ D = B and A ∩ C = ∅ and A ∩ D = ∅, then C=D.

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Q: Given a permutation S={4,5,7}, Find: a.) its inverse b.) even and odd permutations

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Q: Let A = {{1} , Ø}, then O{Ø} C A, Ø CA O{Ø} Z A, Ø C A O{Ø} Z A, Ø L A O{Ø} C A, Ø L A

A: In this question we have to solve a set A.

Q: Suppose |A| = |B|. Prove that |P(A)| = |P(B)|

A: Given that |A|=|B| We need to prove that |P(A)|=|P(B)| Since, |A|=|B|, then there is a bijection ,…

Q: Let A be uncountable and B be countable. Prove that A\B is uncountable.

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Q: Let 9= {[a,b]: a,bER}, then 9 is Borel algebra

A: TRUE ;

Q: Prove that there is no permutation a such that a'(12) a = (34) (15). %3D

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Q: Q1// Let Ai be even permutation in Sn. Then O(An)=n!/2 إضافة ملف

A: Question 1) Sn denotes the set of all permutations of n objects. Then, |Sn| = n! &, An denotes…

Q: 4. Prove: Let a, b e Z*. If a|b and bla, then a = b or a = -b.

A: If a/b and b/a then prove that a=b , a=-b.

Q: In an A.P show that an + a m-n 2a, m+n

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Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.