Let S be a set defined as S={x∈R∣−1≤x≤1}. Determine whether S is open, closed, both, or neither in the standard topology on R.

Q: 1. Give an example of a set S with a relation ~ on S such that ~ is symmetric and transitive but not…

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Q: Consider the following sets in the dictionarv order. Determine which of these sets are linear…

A: We know that Linear continuum is a linearly ordered set S of more than one element that is densely…

Q: let x be an infinite set and let ta be a topology on x in which all infinite subsets of x are open…

A: Given that X be an infinite set and let τ be a topology on X in which all infinite subsets of X are…

Q: A set X and a topology on X which is not Hausdorff.

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Q: Let R be a transitive relation on set A. Define Q to be {(r Prove or disprove that (RUR®)n (QU Qº)…

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Q: a) Show that the set S = {(1, 2, 3), (0, 1, 2), (– 2, 0, 1)} spans R³. b) Determine whether the set…

A: See the attachment.

Q: Consider the set of real numbers, R. Let A = {x∈R:0≤x≤2} be a subset of R. Is the set A open,…

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Q: Determine whether or not the relation R on the set of all people is reflexive, netric,…

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Q: Suppose R and S are symmetric relations on a set A. Prove that RoS is symmetric iff RoS = So R.

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Q: 3) Let X be a set with > 3 elements. Show that there exist bijective functions f :X → X and g : X +…

A: Let x=1,2,3 then, f=123132g=123321 `

Q: Find an example of a set A with |A| = 3 which contains only other sets and has the following…

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Q: Jive R² which are example of two subsets homeomorphic. not C specify the set explicitly) g

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Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

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Q: Find a topology in which the point x = 5 is an open set.

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Q: let A, B ⊂ R, and suppose A and B are closed. Prove that A ∩B is closed. Hint: You might try showing…

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Q: Let Z = {1,2, 3, ...,50}. a. How many relations on the set Z are both reflexive and antisymmetric?…

A: Here given set Z=1, 2, 3, ....., 50

Q: A set F is called an Fo-set (or an Fo) if it is a countable union of closed sets. Likewise, a set G…

A: a) Take An=( -1/n , 1+ 1/n ) , which is open  Then taking intersection from n=1 to infinity we will…

Q: a = b a~b,

A: Here, a¯ represents equivalence class of a, which is defined as, a¯=x∈A/a~x