( Prove that the set {a E R2: a2 ... Prove that the set {a E R2: a2 not equal to 6} is open.

Q: Let A and B be sets. Prove that A U B = An B if and only if A = B

A: First show if A∪B=A∩B then A=B, then show that if A=B then we have A∪B=A∩B.

Q: 3. Find an example of two nonempty sets A and B for which A x B = Bx A is true.

A: To find an example of two nonempty sets A and B for which AXB=BXA

Q: Prove that the set {a E R: a2 not equal to 6} is open.

A:

Q: Let a1 = [-2  h], and a2 = [10  -20] The set {a1,a2} will span R3 unless h

A:

Q: Show that if a < c< b, then fave (a, b) = () fave l(a, c) + C - a | favg [c, b]

A: Solution...

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

A: Consider the given set S. S=1,x2,3+x2 Now, make the matrix from the given set. 103000011

Q: d) Give a recursive definition of the sequence {a„}, n = 1, 2, 3... %3D n if a,= 2 + 1; and a15 e)…

A: d)  Here, given that an=2+1=3 So, for all n, an=3 which is the recursive definition.

Q: Which of the following set is empty a. {x: x = x} b. { x : x != x 2 } c. { x : x ! = x } d. {x:…

A: Empty set:            The empty set is the unique set having no elements. and its cardinality is…

Q: Let A:= {2} and B:= {2,3,5}. Give the Cartesian product of sets A and B, furthermore the power set P…

A:

Q: Does A − (B − C) = (A − B) ∪ C hold for all sets A, B, and C? Prove or give a counterexample for…

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Q: If P(A₁) > 0 and if A2, A3, A4,... are mutually disjoint sets, show that P(A2 U A3 U ... |A₁) =…

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Q: (19) Prove or disprove ∀ sets A, B, and C: (A – B) − C = (A – C) – (B – C).

A: We will use the Venn diagram show the proof: Left side:

Q: Assume that A, B, C and D are sets. Verify that (A x B) U (C x D) ≤ (AUC) x (BUD).

A:

Q: Give an example of four sets A, B, C and D such that (AUC) x (BUD) ≤ (A x B) U (C x D).

A: The set A is a subset of B if every element of A is in B

Q: d) The set E = (-1,0) U (0, 1] is an open set. %3D

A: Given set is E = (-1, 0) ∪ (0, 1]

Q: (2b) Let A and B be sets. Prove that A – B is disjoint from A n BC LA O BE

A: The set A∩B is defined as A∩B=x|x∈A and x∈B The set A-B is defined as A-B=x|x∈A and x∉B Two sets are…

Q: Suppose A={-1, 0, 4}.Find a set B such that A is equivalent to B.

A: Two sets A and B are said to be equivalent to each other if both the sets A and B have same number…

Q: (ANB) –C= (A –C)NB.

A:

Q: Which of the following are difference sets? (ii) {0, 1, 3, 5} in Z13,

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Q: Use the roster method to describe the elements of the following sets. ([-4,6) n (-3, 11)) nZ =…

A: The given sets,The objective is to describe the elements in the given set by roster form.

Q: VII.1. Describe the set (R × Z) ∩ (Z × R)

A: We have to describe the set (R × Z) ∩ (Z × R).

Q: Prove: bd (s) = 5 TR-S bd (s) is a closed set

A:

Q: prove: for any two sets X and Y, #(X x Y) = #(X) • #(Y)

A: To prove it equals the product of the cardinality of both the sets.

Q: 2. Prove Lemma: For all sets A, B, and C, A x (BnC) = (A × B)n (A × C)

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Q: Are the sets Z x Z; ((x.y): x*y?) closed in (R , | |) ? What about the sets (0,1]U(2,3); [0, 00) in

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Q: Does the set {2+x²,1+x.x²} span P₂? Yes No

A: Given the set is 2+x2,1+x,x2

Q: c) [5 marks] Let A, B be any two sets. Prove that if A C B, then An B = 4.

A: * The intersection of complementary sets is empty.

Q: Which of the following sets spans R°? None of these {(-3,0,2)} O {(-3,0,2),(5,1,0),(0,1,1)}…

A: The solution is given as

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: Write out the elements of the set 2X where X= = {1,{0}, Ø, {2,3}}.

A:

Q: Let A and B be sets. Prove that (A − B) ∩ (B − A) = ∅.

A: