Let C be a nonempty subset of a bounded set D. Why does inf C and sup C exist? Show that (c) inf D ≤ inf C and (d) sup C < sup D.

Q: (c) Let X, Y and Z be sets. Prove that (X\Y)n (X \ Z) = X \ (YUZ).

A: We prove the given statement by using set theory.

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

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Q: Let S be the set S={(x+y) l 2x+3y>=(greater than or equal to) 12} What would be Sc  (S complement)…

A: The given set is: S=x,y∈ℝ2 | 2x+3y ≥12 Cleary S⊂ℝ2.

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: etermine whether the set of x-2x²,2+5x-x²x+x²} is linerly independent in P

A: First we use definition of linearly independent Set and we check given vectors are linearly…

Q: Prove that P(A) NP(B) = P(ANB), where A and B are arbitrary sets

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Q: E. Show that both R and R² may be covered by countably many open balls.

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Q: Let P(x; y) : x plays in y A(x) : x is athletic. S(x) : x is smart.  E(y) : y is in English…

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Q: The HAIR Company offers two brand of shampoo:Brand H and Brand S.There where 100 people who were…

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Q: Suppose F and G are families of sets. Prove that (UF)n (UG) ≤ U(F N G) iff VA EFVBE G(An BCU (FNG)).

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Q: (i) If A and B are two sets such that ACB, then m* (A) ≤m* (B).

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Q: Show that the interval [0,1] is a closed set in R

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Q: Let A = {1,3} and B = {2,4}. (a) Draw a picture of the set A × B. (b) Compute P(A × B).

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

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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: Find the closure of each of the following sets: mod T d (a) (3,5) U {6} (b) (-∞, 0) U (0, 1)

A: Given: (a) 3, 5∪6     b -∞, 0∪0, 1 To find: Closure of the given set.

Q: If X and Y are denumerable sets, then X - Y is countable.

A: Given that X and Y are denumerable sets. Definition: A set is denumerable if it can be put into a…

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

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Q: Show that both R and R² may be covered by countably many open balls.

A: To show :  Both ℝ and ℝ2 may be covered by countably many open balls.