11. Let R and R' be two rings. A mapping f: R→R' is called an antihomomorphism, if f(x+y)=f(x) + f(y) and f(xy)=f(y)f(x) x, y € R. Let f, g be two antihomomorphisms of a ring R into R. Prove that fg: R R is a homomorphism.

Q: Q. No. 1: Prove the following using propositional logic: Prove that -[rv(q^(¬r → →p )] = ¬r^(pv¬q)…

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Q: It says "Then there must exists some Aj which covers 0". Why 0? I don't understand. Please explain

A: You can take any point x in [0,1].  There exist a Aj which covers x.

Q: Use double set inclusion to prove that dep. PANB) = P(A)n P(B).

A: The problem is related with power set P of sets. Here, we have to show that P A ∩ B = P A ∩ PB using…

Q: Prove that the set of positive integers is equal to the set {1,2,3,...}. (Using Axioms)

A: This is a problem of set theory and cardinality.

Q: The cardinality of { Ø, 1, 2, 2, {1, 2}} is

A: We need to find the cardinality of { ϕ, 1,2,2,{1,2}}.

Q: 26) Let "" denote the truth-functional connective "exclusive OR." and let "A,1 "v, " and -" denote,…

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Q: Complete congruence statement AJIK = A

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Q: onsider (G,*) where * is defined as a*b3Da is the set of all non-zero real numbers

A: Let  G=ℝ-0For a,b∈G⇒a*b=ab2a,b∈ℝ-0⇒ab2∈ℝ-0a*b∈G   The closure prosperity For…

Q: [2, 5]U[3, 7] [2,7]

A: Let A=[2,5]and B=[3,7]and we have to prove that AUB=[2,7]

Q: * DeMoivre's Theorem 4 (4(os 30 + ) i sin 30

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Q: Show that proof by exhaustion is valid using propositional logic

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Q: За, n > 0, is n+1 ) One solution of the recursion relation an+2 = a) a2k+1 3*a1 k>0 2*k! b) a2k+1…

A: We have to find the one solution of recursion relation.

Q: 4(cos 40+ isinug) • 4 (us 110 + isin 5) leave ansuers in trigonometric form

A: r(cos θ+i sinθ)= r ei θ

Q: Prove: PAQ -QAP using inference rules

A: To Prove- P∧Q - Q∧P by using Inference Rules.     Rules Used-  Simplification: P∧Q∴P Conjunction:…

Q: What equivalence/law was applied? [-p v (p A -q)] v [r v (q A -r)] = [(-p v p) ^ (-p v ¬q)] v [(r v…

A: We will find out the required law.

Q: Z+ is the set of positive integers. Let A = {1,2,3), B = {n ∈ Z+ | n is even}, C = {n ∈ Z+ | n is…

A: concept Used: C∩D is the set  containing all elements of C that also in the set D C∪D is the set…

Q: I need help in simplifying this using the Logical Equivalence Rules (De Morgan's Laws etc.) [(~p ^…

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Q: Let the operation on N defined by a*b=a-b, Va,beN. Then, find the property which satisfies by the…

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Q: Construct a truth table for the following Boolean function showing all columns: (x + y')(x' + z')(y'…

A: 1'=0 0'=1 1+1=1+0=0+1=1•1=1 0+0=1•0=0•1=0•0=0

Q: Prove that the interval (0, 1) has cardinality aleph-one.

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Q: and proof of an existence theorem that can be proved by “nonconstructive proof” and provide the…

A: The proof is the proof of Intermediate value theorem.

Q: Question 3 This ~(a → b) = a ^ ~b is a logical equivalence. True False

A: We need to see if ~a→b≡a∧~b is logically equivalent. To see if they are logically equivalent we need…

Q: ` Prove that positive real numbers a, b, c, d, b 2 + > 3.. a b+zc+3d +c+zd+za+d+za+36 d a+zh+30

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Q: Prove that the following statement is a tautology (~q^(p⇒q)) ⇒~p

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Q: Show the following proposition using a proof by contraiction  Proposition. I a,b, E Z, then a^2-4b-2…

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Q: If this is a theorem, derive it in propositional logic. If it is not, show that it is not using a…

A: Given-  ~(M ↔ N) → ~((M & N) v (~M & ~N) To prove it by propositional logic

Q: Assume that A = {1, 2, 3, 5, 8, 7} B = {5, 3, 4} C = {2, 3, 5, 6} and that U is the universal set of…

A: It is basic set theory problem

Q: Prove using the sequent calculus that (pAq) V-p V ¬g is a tautology.

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Q: Prove that all integer solutions form an enumerable set without using Fermat Theorem

A: Given: Set of all integer solutions To prove: Set of all integer solutions form an enumerable set.

Q: Determine in set notation where Continuas nuuas S! x^~X + E =X)

A: Given , f(x) = 3 + x2 - √x We have to determine where the function is continuous

Q: What postulate or theorem justifies the statement AAVB ANVK?

A: Given- △AVB~△NVK To find- What postulate or theorem justify the above statement.

Q: Prove that each of the following pairs of sets have the same cardinality. (a) We have the sets X and…

A: In set theory, one important concept is that of cardinality, which is the measure of the "size" of a…

Q: Prove the empty set hasn't supremum and infimum

A: Lower Bound: A lower bound of a subset A of a partially ordered set (X,⩽) is an element a∈X such…

Q: Let F(x, y) be the statement x can fool y, where the domain consists of all people in the world. Use…

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Q: Please poove 18 ŏ bound of (३5) fश्ठ also proove ppes an sup13,5)%35

A: (.) Upper bound:    "A number that is greater than or equal to any number in a non-empty subset of ℝ…

Q: Construct the truth tables of: (a) -(p^q) (b) p^(-q) (c) (p^q) Ar (d) (p^q) v (q^r) V (r/ (e) -p V-q…

A: Constructing the truth table for the given by taking combinations of truth and false values of the…

Q: Question 3. Show that [p ^ (p^q)] V (p→q) is a tautology using logical equivalences. for a list of…

A: