4. Prove that if > is a tautology, then (& → ) is a tautology for any & Є L. 5. Prove that if antradiction then ( 4 EL.
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- Label each of the following statement as either true or false. 9. If ab and ba then a=b.[Type here] 7. Let be the set of all ordered pairs of integers and . Equality, addition, and multiplication are defined as follows: if and only if and in , Given that is a ring, determine whether is commutative and whether has a unity. Justify your decisions. [Type here]1. Let L(x, y) be the statement "x loves y" where the domain for both variables x and y comprises people's names. Use quantifiers to express the statements below. As an example, the statement "Somebody loves Jim" is expressed by 3x L(x, Jim). a. Everyone loves someone. b. Someone loves everyone. c. d. e. f. g. No one loves everyone. Someone loves no one. Jane loves no one. No one loves Jane. John loves no one except for Kim.
- 3. Let M(x, y) be “x has e-mailed y" and C(x,y) be “x has called y," where the domain consists of all students. Express each of the following sentences as a quantified statement. a. There is a student who has either e-mailed or called every student. b. There is a student who has e-mailed exactly one student. c. There are two different students such that the first student has called the second but the second student has not called the first.3. Give an example of an open sentence P(x, y) so that the quantified statement ( Vxy, P(x, y)) is True, while the quantified statement (yx, P(x, y)) is False, and explain why this is so.e) Let A(x), C(x), S(x) and GP(x) be the statements "x is an animal," "x is a cat," "x is small," and "x is a good pet," respectively. Express each of these quantifications in English where the universe consists of all animals. i. Vx (C(x) → A(x)) ii. -3x(C(x) ^ ¬ S(x)) i. Vx (C(x) S(x) A A(x)) iv. Vx (S(x) A A(x)→ GP(x))
