Suppose f: A C and g: B → C. Prove that if A and B are disjoint, then f U g: A U B → C. Prove that f u g: A U B → C exercise 7 for the meaning of the notation used here.) C iff f ↑ (A n B) = g ↑ (A n B). (See

Q: If ab is algebraic over F and b ≠ 0, prove that a is algebraic over F(b).

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Q: Answer with True or False. 1. The functions from Z to Z is onto if f(n)=n−1. 2. These function…

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Q: Prove the following: If ā and b are in R', then (5 -a)×(5 +a)=-2(āxb).

A: Given a⇀ and b⇀ are in R3. To prove :  b⇀-a⇀×b⇀+a⇀=-2a⇀×b⇀

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A: Given, f: R→R be                       f(x)=x1+|x|

Q: be a field and let c,d e F. Show that c · (-d) = -(c d).

A: Associative Property of Field F for a,b,c∈F a·b·c=a·b·c

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Q: The set AcR is dense in Rif .A A =R .B A°=R .C (Ā)° = Ø .D none of above

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Q: Let S be the set S = {a+bk : a,b ∈ R}, where k is a formal symbol. Define addition and…

A: Let xy=ac+bd+ad+bck Then first we find multiplicative identity. Let x=a+bky=c+dk Then…

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Q: Let E be an extension of F and let a and b belong to E. Prove thatF(a, b) = F(a)(b) = F(b)(a).

A: Given Data E be the extension of F, and a & b belong E.   The E is an extension of F, and…

Q: Prove that the class {AC – B, BC – A, AB − C} is pairwise disjoint.

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Q: (d) Using Part (c) and continuity of g, prove that g(x) = mx + l for some m, l e R. (e) Using…

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Q: Let K' be and K' be to divisions of [a,b]. Show that  L(f,K') ≤ U(f,K'')

A: We need to prove the following : If f is a bounded function on [a, b], and if P and Q are partitions…

Q: The set AcRis dense in Rif .A A =R .B A°=R .C (A)° = Ø .D none of above

A: Option (A) is correct.

Q: Define two operations + and * on A = Z × Z as follows: for (a, b) and (c, d) in A: ɔy (a, b)+(c, d)…

A: As per our guidelines we are allowed to solve one question at a time. Kindly repost the rest…

Q: Let f: [a,b]-R be afunction and let c<0, then for any Ac[a,b] sup cf=c inf f

A: We will prove it

Q: Define two operations + and * on A = Z × Z as follows: for (a, b) and (c, d) in A: ɔy (a, b)+(c, d)…

A: As per guidelines, we will solve first three subparts: Define two operations + and * on A=ℤ×ℤ as…

Q: Show that given relation is Symmetric For X, y e R say x is congruent to y modulo Z if x - y is an…

A: This question is related to number theory.

Q: Let A, B, C, and D be sets.

A: First, (AxC)U(BXD) ⸦(AUC) x(BUD)Take left hand side first , let  (x,y) ∈(AxC) U(BxD)=&gt; (x,y)…

Q: Suppose f: A → B, f is one-to-one and onto, and Y c B. Show that the inverse image of Y under f and…

A: The function f is one-one and onto.   The function f is one-one if two different element in A…

Q: 1]If 9:T>V and f:ST and gof IS injective, prove that both f añd 9 are injective.

A: Injective function:- for every x1=x2 implies that f(x1)= f(x2) .or vice versa.

Q: Define f:Z* x Z* →z* by the formula f(m, n)= 4"6" for all (m, n)Ɛ Z* × Z*. (a). Is f one-to-one?…

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Q: Suppose f(x) =. Evaluate: A - In (2) :- 4 3 In (2)

A: We will use properties of logs to solve it.

Q: a field and let c,d ∈ F. Show that c⋅(−d) = −(c⋅d).

A: Associative Property of Field F for a,b,c∈F a·b·c=a·b·c

Q: Consider f:Z →S, such that f is homomorphism and f(1)= (1426)(257), then f is one-to-one. %3D Select…

A: The given statement is false . one to one mapping: A mapping  f:A→B  is said to be one to one if for…

Q: O Prove that the composition óf önlo

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

Q: Show that given relation is transitive For X, y € R say x is congruent to y modulo Z if x – y is an…

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Q: If f, g E R[a,b] with f < g on [a,b], prove that g.

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Q: Let f:[a,b]→R be afunction and let c<0, then for any A C[a,b] sup cf =c sup f ylgn

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Q: Use the functions f and g in C[-1, 1] to find (f, g), ||f||, ||g||, and d(f, g) for the inner…

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Q: Which of the following is isomorphism? * f:(Z, +) → (R*, .) where f(x) = 2* O this option f: (Z, +)…

A: A mapping is said to be isomorphism if it is one-one, onto and homomorphism. A mapping f: X→Y is…

Q: Question 3. Consider the relation R(ABCD) with AB as a candidate key, choose the FD so that R should…

A: Given the data:

Q: Prove: f € R(a) if and only if for each ɛ > 0, there exists a partition of [a, b ] , say go, such…

A: f∈Rα if and only if for each ε&gt;0,there exists a partition of a, b,say P, such that UP, f, α-LP,…

Q: Given that * :Zx Z Z is defined by m * n = m" for m, n e Z, then is an operation on the set of…

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Q: Show that if f : A → B and g : B → C are one-one, then gof : A → C is also one-one.

A: To show that the composition of two one to one function is also one to one.

Q: Let f, g, h : N→ R²0. Prove or disprove that if f + g € O(h), then ƒ e O(h) and g E O(h).

A: Given f,g,h:ℕ→ℝ≥0 such that f+g∈Oh. We know that f∈Oh, if h is strictly positive for all large…

Q: 4. Define two operations + and * on A = Z x Z as follows: for (a, b) and (c, d) in A: (a, b)+(c, d)…

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Q: the only homomorphism f : S, to Z,, is the trivial one, Select one: O True O False

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Q: 4. Define two operations + and * on A = Z x Z as follows: for (a, b) and (c, d) in A: (a, b)+(c, d)…

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Q: Let E be an extension of F, and let a,b be elements of F. Prove that F(a,b) = F(a)(b)=F(b)(a)

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Q: If f,g continuous on [a,b] and f(a) g(b). Show that there exists c ∈ (a,b) such that f(c) = g(c)

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Q: Prove: If f∈ R(a) on [a, b] and if a < c < b, then f∈ R(a) on [a, c] and on [c, b] and ∫ab f da =…

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Q: Let f : A B with C, C1, C2 C A and DC B. (a) Show that if f is INJECTIVE (ONE-TO-ONE) then ff(C)] =…

A: "Since you have posted multiple questions, we will answer the first question.  If you want the…

Q: Suppose that E is an extension of F and a, b ∈ E. If a is algebraic over F of degree m, and b is…

A: Given that E is an extension of F and a, b ∈ E. If a is algebraic over F of degree m, and b is…

Q: Suppose you have the following relation: R: A-A where A = (a, b, c, d, e) %3D R= ( (a, b), (a, e),…

A: Given relation is R:A-&gt;A where A={a,b,c,d,e}R={(a,b)  , (a,e) , (b,e) }determine whether it is…

Q: Suppose f : Z → Z is defined by f(m, n) = m – n and g : Z→ Z is defined by g(n) = "i. (a) Is f…

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Q: Suppose that E is an alphabet, and that f :E* → E* has the property that f(ơ) = o for every o e E an…

A: Given information: Given that Σ is an alphabet. The given function is f.

Q: "for all n,m ∈Z \ {0} and g ∈ N,  gcd (n,m)= g then there are a,b ∈ Z with an+bm= g" is the…

A: We need to see if the converse of the statement "gcdn,m=g then there are a,b∈Z with an+bm=g" is true…

Q: Let a, b, c, d e R with a < b and c < d. Show that [a, b) is homeo- morphic to (c, d].

A: