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Question: fundamental mathematics discrete math proofs help please 1 express each...

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FUNDAMENTAL MATHEMATICS/ DISCRETE MATH/ PROOFS

Help please!!

1. Express each of the following sentences in the ”If ..., then ...” form. There are many possible answers.

a) You must eat dinner if you want to grow.

b) Being a multiple of 12 is a sufficient condition for a number to be even.

c) It is necessary for you to pass your exams in order to obtain a degree.

d) A triangle is equilateral only if all its sides have the same length.

2. Consider the sentences:

(i) If I am to get a new bicycle, I must do my homework.

(ii) The United States must play more soccer if it is to win the World Cup.

a) Rewrite the above sentences using the words ’necessary’ or ’sufficient’.

b) What are the converses of the sentences in a)?

c) What are the contrapositives of the sentences in a)?

d) What are the negations of the sentences in a)?

3. Use De Morgan’s law to prove that P ⇒ Q is logically equivalent to ¬P ∨ Q.

4. (Extra) Suppose that P and Q are propositions. Argue that for any of the 16 possible truth tables

there exists an expression E, formed with P, Q, ∨, ∧ and ¬ realizing this truth table. Can you generalize this result in any way?

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