Homework Answers
Proper subset always means A⊂B;A≠B, so of course '∅'is a proper subset of any set, except for itself,where A is the subset of B.
This means that,
In almost every case Empty set is a proper subset of a set if the main set is not a empty set except
When the very main set is an Empty set .only then empty set is called an improper subset.
We prove this by the method of contradiction as follows
Add Answer to:
Translate "the empty set is not a proper subset of every set" into a predicate logic expression and then use 1...
At the end of lecture 5 (see the recordings) we saw how to use predicate logic to prove that syll...
Example of the syllogism to set-theoretic translation
At the end of lecture 5 (see the recordings) we saw how to use predicate logic to prove that syllogism types are valid. Prove that the following syllogism is valid by following the steps below My teapot is purple My teapot holds water There exist purple things that hold water a) Translate the syllogism into set-theoretic notation b) Translate your set-theoretic notation into the notation of predicate logic c) Give a proof that...
(d) Translate the following statement into predicate logic: “Every function f :R → R can be...
(d) Translate the following statement into predicate logic: “Every function f :R → R can be written as the sum of an even function and an odd function.” You can use the notation fi + f2 to represent the sum of functions fı and f2, and the notation f1 = f2 to represent the fact that fi and f2 are equal. 2n izo (e) Let n € N, and 20, 21, ..., Q2n E R. Let f: R + R...
I posted these question before but the answers turned out wrong, please help.(Monadic predicate logic) The...
I posted these question before but the answers turned out wrong, please help.(Monadic predicate logic) The ones required are = ( tilde ~ for negation, dot • for conjuction, horseshoe ⊃ for material implication( the conditional ), vel ∨ for disjunction, triple bar ≡ for biconditional ) Please use these symbols. translate the following English sentences into Predicate Logic: 1. All philosophers are scientists. (Px, Sx) 2. Some mathematicians are philosophers. (Mx, Px) 3. No chess players are video gamers....
it is about the classical logic in the subject of formal method： the question is shown...
it is about the classical logic in the subject of formal method：
the question is shown as the picture
Question 1: Classical Logic [25 marks)
a) Answer the following questions briefly but precisely.
i. State what it means for an argument to be valid in Predicate
Logic. [3 marks
ii. Suppose you use resolution to prove that KB = a. Does this
mean that a is valid? And why? [3 marks
b) Consider the following three English sentences: Sl: If...
DISCRETE MATHEMATIC For question 1, Use mathematical induction to prove the statements are correct for n...
DISCRETE MATHEMATIC For question 1, Use mathematical induction to prove the statements are correct for n ∈ Z+(set of positive integers). 1. Prove that for n ≥ 1 1 + 8 + 15 + ... + (7n - 6) = [n(7n - 5)]/2 For question 2, Use a direct proof, proof by contraposition or proof by contradiction. 2. Let m, n ≥ 0 be integers. Prove that if m + n ≥ 59 then (m ≥ 30 or n ≥...
5- Recall that a set KCR is said to be compact if every open cover for...
5- Recall that a set KCR is said to be compact if every open cover for K has a finite subcover 5-1) Use the above definition to prove that if A and B are two compact subsets of R then AUB is compact induction to show that a finite union of compact subsets of R is compact. 5-2) Now use 5-3) Let A be a nonempty finite subset of R. Prove that A is compact 5-4) Give an example of...
1. (a) Let d be a metric on a non-empty set X. Prove that each of...
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
y, July AM 1. What does it mean for a sequence {a} to converge to a...
y, July AM 1. What does it mean for a sequence {a} to converge to a € R? State the definition (-1)+1 What about sequences that don't converge? Read the following proof by contradiction, and then complete Practice Question 6. Claim: {(-1)"} does not converge to any real number a. Proof: Assume that the sequence converges; that is, assume that there is an a E R such that lim,-(-1)" = a. Then, using & = 1, from the definition of...
Question 1: Let R be the set of real numbers and let 2R be the set...
Question 1: Let R be the set of real numbers and let 2R be the set of all subsets of the real numbers. Prove that 2 cannot be in one-to-one correspondence with R. Proof: Suppose 2 is in one-to-one correspondence with R. Then by definition of one- to-one correspondence there is a 1-to-1 and onto function B:R 2. Therefore, for each x in R, ?(x) is a function from R to {0, 1]. Moreover, since ? is onto, for every...
Please help me solve 3,4,5 3- For all n € N, let an = 1. Let...
Please help me solve 3,4,5
3- For all n € N, let an = 1. Let S = {an in€ N}. 3-1) Use the fact that lim - = 0 and the result of Exercise 1 to show that 0 ES'. Ron 3-2) Use the result of Exercise 2 to show that S = {0}. 4- Prove that 4-1) N' = 0. 4-2) Q =R. 5- Recall that a set KCR is said to be compact if every open cover...
Example of the syllogism to set-theoretic translation
At the end of lecture 5 (see the recordings) we saw how to use predicate logic to prove that syllogism types are valid. Prove that the following syllogism is valid by following the steps below My teapot is purple My teapot holds water There exist purple things that hold water a) Translate the syllogism into set-theoretic notation b) Translate your set-theoretic notation into the notation of predicate logic c) Give a proof that...
(d) Translate the following statement into predicate logic: “Every function f :R → R can be written as the sum of an even function and an odd function.” You can use the notation fi + f2 to represent the sum of functions fı and f2, and the notation f1 = f2 to represent the fact that fi and f2 are equal. 2n izo (e) Let n € N, and 20, 21, ..., Q2n E R. Let f: R + R...
it is about the classical logic in the subject of formal method：
the question is shown as the picture
Question 1: Classical Logic [25 marks)
a) Answer the following questions briefly but precisely.
i. State what it means for an argument to be valid in Predicate
Logic. [3 marks
ii. Suppose you use resolution to prove that KB = a. Does this
mean that a is valid? And why? [3 marks
b) Consider the following three English sentences: Sl: If...
5- Recall that a set KCR is said to be compact if every open cover for K has a finite subcover 5-1) Use the above definition to prove that if A and B are two compact subsets of R then AUB is compact induction to show that a finite union of compact subsets of R is compact. 5-2) Now use 5-3) Let A be a nonempty finite subset of R. Prove that A is compact 5-4) Give an example of...
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
y, July AM 1. What does it mean for a sequence {a} to converge to a € R? State the definition (-1)+1 What about sequences that don't converge? Read the following proof by contradiction, and then complete Practice Question 6. Claim: {(-1)"} does not converge to any real number a. Proof: Assume that the sequence converges; that is, assume that there is an a E R such that lim,-(-1)" = a. Then, using & = 1, from the definition of...
Question 1: Let R be the set of real numbers and let 2R be the set of all subsets of the real numbers. Prove that 2 cannot be in one-to-one correspondence with R. Proof: Suppose 2 is in one-to-one correspondence with R. Then by definition of one- to-one correspondence there is a 1-to-1 and onto function B:R 2. Therefore, for each x in R, ?(x) is a function from R to {0, 1]. Moreover, since ? is onto, for every...
Please help me solve 3,4,5
3- For all n € N, let an = 1. Let S = {an in€ N}. 3-1) Use the fact that lim - = 0 and the result of Exercise 1 to show that 0 ES'. Ron 3-2) Use the result of Exercise 2 to show that S = {0}. 4- Prove that 4-1) N' = 0. 4-2) Q =R. 5- Recall that a set KCR is said to be compact if every open cover...
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