7. Prove: If A is any set of 9 integers between 1 and 50, then there exist two different subsets XC A and Y C A for which the sum of elements in X equals the sum of elements in Y.

Q: Suppose V is a set and that n(V) = 10. How many subsets of V contain exactly 3 members?

A: Number of ways in which 3 elements of V can be clubbed to form a subset = 10C3…

Q: Show that, for given integers a, b, c, if gcd(a, c) = gcd (b, c) = 1, then gcd (ab, c) = 1

A: Given a,b,c∈Z+ And gcda,c=1 and gcdb,c=1 We have to show that gcdab,c=1 Since, gcda,c=1 and gcdb,c=1…

Q: Prove that there exixts integers m and n such that 15 m + 12n = 3

A: Given equation is 15m+12n=3

Q: Prove or disprove that given a set of 125 integers, one can choose a subset of S of 16 of them so…

A: Given we have set which contains 125 different numbers. When we divide each number in set by 8 we…

Q: Suppose X is a set with 84 elements that is partitioned into four subsets: X1, X2, X3, and X4. If X1…

A: Given total number of elements = 84 Let  number of elements in X4 be a As X1 has twice as many…

Q: Using Pigeonhole principle to prove, if there exists 6 integers (between 1 and 10) in set S, then…

A:

Q: 2. Show that Z has the same cardinality as the set O of odd integers.

A: Given below the detailed solution

Q: 6. Prove that if x, y, and z are integers and x + y + z is odd, then at least one of x, y, and z is…

A: Given: x , y , and z are integers and x + y + z is odd.  We have to show that at least one of x , y…

Q: 4. Use set identities or different proof method to show that: (A U B) n (B U A°) = B

A:

Q: Let $n$ be a positive integer. Among C(2n,0),C(2n,1),..., C(2n,2n), C(2n,n) is the largest. True…

A:

Q: 11. Establish the following relation between Fibonacci and Lucas numbers: L,=u, + 2u,-1

A: Introduction: We know that the relation of Fibocanci number and Lucas numbers is given by  F0=0,…

Q: 4. Is the set Q² x {a,b,c} countable or uncountable? You must justify your answer with a proof.

A: Given : ℚ2×a , b ,c  To Prove : ℚ2×a , b ,c  countable or uncountable

Q: 1.(a) Do there exist integers a, b, c such that gcd(a, b) = 30, gcd(b, c) = 24, gcd(a, c) = 54?…

A: (a) gcd(a, b) = 30       gcd(b, c) = 24       gcd(a, c) = 54      => 30 | a, 54 | a           24…

Q: 4. Consider a problem 7 defined as; T: Given a set of n positive integers A = {a1, a2, · ·· , an},…

A: To display that any difficulty A is NP- complete, 4 matters are critical to be appeared. The first…

Q: 6.29. Prove that if A1, A2,.., An are any n > 2 sets, then ..... A1n A2 n..n An = A1U A2 U...U An.…

A: Prove that A1∩A2∩A3∩...........∩An ¯ =A1¯∪A2¯∪A3¯∪..........∪An¯

Q: 4. Show that if n is a positive integer, then Σ {a₁,...,ak}{1,2,...,n} (Here the sum is over all…

A: Given, If n is a positive integer, then we have to show that ∑a1,.....ak⊆1,2,....,n 1a1a2....ak=n

Q: If m, n, p, q are integers, n > 0, q > 0, and r = m/n = p/q, prove that

A:

Q: Suppose if p and q are odd integers with p!= q (p is not equal to q). Show that there is a unique…

A:

Q: *prove that for . P+2+3+... +n° = 2 n=!,2,3,

A:

Q: 2. Use a Venn diagram to illustrate the subset of odd integers in the set of all positive integers…

A:

Q: Suppose that a, b, and c are integers. Prove that if a2 +b2 =c2 then a or b is even.

A:

Q: Suppose A1, A2, ..., 430 are thirty sets each having 5 elements and B₁, B2, ..., B₁ are n sets each…

A:

Q: (b) How many minimal elements are there in the ordered set (X,|), for X = {4k + 2: k > 2}? Let us…

A: Given the poset ordered set X,| for X=4k+2:k≥2. To find minimal elements, we must find a∈X such that…

Q: 2.4 a. Find all the cluster points of the set Q of all rational numbers, and justify your…

A:

Q: Prove or disprove that all chessboards of these shapes can be completely covered using L-shaped…

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

Q: السؤال 2 2 The set P of positive integers has maximal element صواب ihi

A: Use th Archemedian property of set theory.

Q: Suppose we randomly select a subset of positive integers. How large must the subset be, if we want…

A: According to the given question, if any positive integer n belongs to the required subset then it…

Q: 1. Let x, y, z be integers. Prove that if x(y + z) is odd, then x is odd and at least one of y or z…

A:

Q: Let S = {(x, y) E R X R: y - x = integer} List of four members of [-3]

A: GivenS=x,y∈IR×IR: y-x=Integernow-3=x∈IR: -3-x=Integernow, 1∈IR,2∈IR , 3∈IR and 4∈IRso the four…

Q: 2. Prove thatN is the smallest inductive subset of R:

A: We can prove this by proving property of induction

Q: Let a and b be positive integers and let d = gcd(a, b) and m =lcm(a, b). If t divides both a and b,…

A:

Q: . Consider a problem 7 defined as; T: Given a set of n positive integers A {a1, a2, ·., an}, is…

A: To display that any difficulty A is NP-complete, 4 matters are critical to be appeared. The first…

Q: 1. Prove that if m + n andn +p are even integers, where m, n, and p are integers, then m + p is…

A:

Q: Let X = {a, b, c}, Y = {a, c, b}, and Z = {a, b, b, c, c, c}. What are the elements of X, Y, and Z?…

A: It is known that the ordering of the elements in a set does not matter. If the elements of both sets…

Q: The sets A and B have 6 and 9 elements respectively, such that A is proper subset B , then the total…

A: Given that set A and set B have 6 and 9 respectively. Now that A is a proper subset of B which means…

Q: Let a, b, and c be integers. Prove that if a divides b and a divides c, then a divides (b-c).

A: a, b and c are integers, a divides b and a divides c. Since a divides b, therefore b=ak,  k is any…

Q: How many 5-tuples (x1, x2, x3, x4, X5) of positive integers from 1 to n are there? How many 5-tuples…

A: Since you have asked multiple questions, we will solve the Question Q1 (a) and (b) and (c) for you.…

Q: Show that if A is a countable set, and B CA is infinite, then B is countable. [Hint: How could we…

A: To prove that  If A is a countable set and B is a subset or equals to A is infinite then B is…

Q: Let Σ = {0, 1, +, =} and S = {x = y+z : x,y,z ∈ {0,1}* and x,y,z are binary integers, and x is the…

A: Given that S=x=y+z By using the pumping lemma. Assume that for the purpose of reaching a…

Q: 2. (i) Show that given finitely many countable sets A1, A2,, An, the set A1 x A2 x .. x An is also…

A: A set A is said to be countable if its cardinality is same as the set of natural numbers ℕ. If A and…

Q: Show that the integers m = 3k . 568 and n = 3k . 638, with k E Z>o, satisfy simultaneously $(m) =…

A:

Q: (c) Show that the set {r EN | x² % 4 = 1} is equal to the set {EN | x² % 8 = 1}.

A: Two sets are said to be equal if the number of elements in both the sets are equal. Now, we have one…

Q: Suppose XX is a set with 144 elements that is partitioned into four subsets: X1, X2, X3, and X4. If…

A: To determine the number of elements in each subset in the partition.

Q: Determine whether the set S = {1, x², 9 + x²} spans P2. %3D O S spans P2. O s does not

A: Introduction: Vectors' span, It is the collection of all linear combinations of a number of vectors.…

Q: Prove that if p,q, and r are integers, r|p and r|q, then r^2|pq.

A:

Q: 5. Deduce from (4) that the set {2} 2-: n = 1, 2, 3, ... is closed in R.

A: Given set  S =2 ∪2-1n  :  n=1  2,  3 , . . .

Q: What is the set Z of all integers?

A: The set of all integers is, ℤ=........, -4, -3, -2, -1, 0, 1, 2, 3, 4, ........

Q: If A subut of B. where B 18 a closo d subset of IR show that

A: Given: A is a subset of B, where B is closed subset of ℝp. To show: A¯⊆B.

Q: If S is the set of all positive integers greater that 19, is S finite or infinite?

A:

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.