Question 2. Prove that if AUB= A is true, then An B = B is also true, AND CONVERSELY.
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- Prove each statement using a proof by exhaustion.Consider the following problem: 1. You have a four-gallon jug and a three-gallon jug, neither of which has any markings. Understand what this means. The only reliable observations are: a jug is empty in which case it has zero gallons in it; or, a jug is full, in which case it has its "capacity" gallons in it, and any other conclusions you can logically draw from these states. For example, say you fill the 4-gallon jug, and then fill the 3-gallon jug from it, you can logically conclude that there is 1 gallon left in the 4-gallon jug. Since the jugs have no markings, no other amounts can be transferred from the full 4- gallon jug to the 3-gallon jug. In this example, you cannot transfer 1, or 1.5, or 2, or 2.25, or 3.142 gallons of water from the 4- gallon jug to the 3-gallon jug. 2. There is a faucet from which you can top off the jugs as many times as you wish. Following this, a jug has its "capacity" gallons in it. 3. You can empty one jug into the other, or onto the ground, in which…Let us assume that p and q are two propositions. Using p and q you are asked to form a number of compound proposition as shown in Table 2. Fill up the Truth Table for all the propostions.
- Prove that: Ā+ A. B = Ā + BE. The relationship a=b indicates that a and b are equivalent. The relationship asb indicates that a and b are congruent. You may shortcut this symbol by underlining an equal symbol such as. To find congruence for some b mod n, divide b by n and take the remainder (r) as the answer. Put your answers in terms of mod 2. For example, to determine (72 mod 2) determine the remainder 72/2 = 36 r 0 therefore, 72 mod 2 = 0 mod 2 since 0 is the remainder or 72/2. a. 24 mod 2 = b. 13 mod 2 = c. 6 mod 2 = d. 29 mod 2 F. Consider a 12-hour clock. People can add hours to calculate another time. All values on the face of the clock range from 1:00 to 12:00 regardless of AM or PM. Figure out the following times: a. 1:00 +3 hours b. 5:00+ 7 hours c. 7:00 +8 hours d. 12:00+12 hours e. 6:00+ 6 hoursFor the given statement write (a) the converse, (b) the inverse, and (c) the contrapositive in if...then form. If it ain't broke, don't fix it. (a) Write the converse in if...then form. Choose the correct answer below. O A. If it is not broken, then it needs to be fixed. O B. If it does not need to be fixed, then it is broken. ce O C. If it does not need to be fixed, then it is not broken. O D. If it is broken, then it does not need to be fixed. (b) Write the inverse in if...then form. Choose the correct answer below. Too O A. It needs to be fixed if and only if it is broken. O B. If it is not broken, then it does not need to be fixed. O C. If it is broken, then it needs to be fixed. O D. If it needs to be fixed, then it is not broken. (c) Write the contrapositive in if...then form. Choose the correct answer below. O A. It does not need to be fixed if and only if it is broken. O B. It is broken if and only if it needs to be fixed. C. If it is not broken, then it needs to be fixed. O D.…
- Consider a blood test for some virus. 200 persoons received the test and here is the result: 120 persons who are healthy tested negative. 5 persons who are healthy tested positive. 25 persons who have the virus tested negative. 50 persons who have the virus tested positive. Answer the following questions. If it is applicable, please write a complete writing of hypothesis statement, Rcode, output and your findings. What is the false positive rate of this test? What is the false negative rate of this test?You need to evaluate a set of propositions to solve the truth values of the propositions. You plan on using a technique of testing by contradiction. The propositional sentences are in a series of conditions, conjunctions, and disjunctions. What form would you choose before evaluating the propositions? Group of answer choices Conjunctive Normal Form Arc Consistent Normal Form Third Normal Form Existential Elimination Normal FormUse logical equivalence to get to the solution. Do not use a truth table.
- Use a truth table to demonstrate that these two statements are logically equivalent:((p ∧ ¬q) ∨ (p ∧ q)) ∧ q and (p ∧ q).Question A Locker can be opened by A and B when they are both there, as the locker has two keys, one of which is with A and other with B. C has both the keys, but he is permitted to open the locker only when B is not there. show that locker can be opened always if both A and C are there. Give boolean identify that symbolises this situation.Fill out the truth table and decide if the propositions are logically equivalent. and