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Consider the following sets together with binary operations. Are they user-friendly? Z with binary operation z1 . z2 = 2z1 - 4z2 Is the set closed under the operation? Is the operation commutative ? Is the operation associative ? Is there an identity? If there is an identity element then does every element have an inverse relative to the operation _ Consider R together with x . y = x/y. Ask the same questions as in last example.
Prove or disprove that C~ R as fields.
1. which of the following statement is true a. \$$\\sim (p\\vee q)=\\sim (p\\wedge \\sim q)\$$ b. \$$(p\\vee q)\\wedge (p\\vee r)=p\\vee (q\\wedge r)\$$ c. \$$(p\\wedge q)\\wedge (p\\vee r)=p\\vee (q\\wedge r)\$$ d. \$$(p\\wedge q)\\vee (p\\vee r)=p\\vee (q\\wedge r)\$$ 2. ____ reads “the goods are standard if and only if the goods are expensive” a. \$$\\sim (\\sim p\\wedge \\sim q)\$$ b. \$$\\sim \\sim q\$$ c. \$$p\\leftrightarrow q\$$ d. \$$\\sim p\\wedge q\$$

1. \$$(p\\wedge q)= (q\\wedge p)\$$ and \$$(p\\vee q) = (q\\vee p)\$$ implies an _____ a. Idempotent Laws b. Associative laws c. Distributive Laws d. Commutative Laws 2. given that\$$A=\\begin{pmatrix}1 & 2 & 3\\\\ 4 & 5 & 6 \\end{pmatrix}\$$\nand \$B=\\begin{pmatrix} 1 & 2\\\\ 3& 4\\\\ 5& 6 \\end{pmatrix}\$\n. Find AB a. \$$\\begin{pmatrix} 0 & 12 & 17\\\\ 19 & 26 & 31\\\\ 29 & 40 & 51 \\end{pmatrix}\$$ b. \$$\\begin{pmatrix} 5 & 12 & 15\\\\ 19 & 26 & 31\\\\ 29 & 40 & 51 \\end{pmatrix}\$$ c. \$$\\begin{pmatrix} 0 & 12 & 17\\\\ 19 & 26 & 31\\\\ 20 & 40 & 45 \\end{pmatrix}\$$ d. \$$\\begin{pmatrix} 0 & 12 & 17\\\\ 7 & 10 & 31\\\\ 20 & 40 & 45 \\end{pmatrix}\$$

1. ____is equivalent to \$$(p\\vee q)\$$ a. \$$\\sim (\\sim p\\wedge \\sim q)\$$ b. \$$(\\sim \\sim q)\$$ c. \$$(p\\leftrightarrow q)\$$ d. \$$(\\sim p\\wedge q)\$$ 2. \$$(p\\vee q)\\vee r=p\\vee (q\\vee r)\$$ and \$$(p\\wedge q)\\wedge r = p\\wedge (q\\wedge r)\$$ implies an ___ a. Distributive Laws b. Commutative Laws c. Associative laws d. Idempotent Laws
Let d ∈N , where d ≠1 and d is not divisible by the square of a prime. Prove that N:Z[square root of d] maps N union {0} : N(a+b sqr root d) = |a^2 -db^2| satisfies the following properties for x,y belongs to Z[sqr root d]. 1. N(x) = 0 if x=0. 2 N(xy) = N(x)N(y) 3. N(x) =1 if x is a unit 4. N(x) is prime if x is irreducible.
If p*q = p^2-q^2-2pq. Find the inverse of p under the operation.
State and prove generalized commutative law in a commutative semigroup
Q. Find the dimension of the subspace of R4 that is span of the vectors (█(1&brvbar;(-1)@0@1)), (█(2&brvbar;1@1@1)),(█(0&brvbar;0@0@0)),(█(1&brvbar;1@-2@-5)) Q. Choose the correct answer. Q. Let b and c are elements in a group G and e is identity element of G. If b5=c3=e,then inverse of bcb2 is a. b2cb b. b3c2b4 d. b2c2b4
Let N &isin;d , where d &ne;1 and d is not divisible by the square of a prime. Prove that N:Z[square root of d] maps N union {0} : N(a+b sqr root d) = |a^2 -db^2| satisfies the following properties for x,y belongs to Z[sqr root d]. 1. N(x) = 0 if x=0. 2 N(xy) = N(x)N(y) 3. N(x) =1 if x is a unit 4. N(x) is prime if x is irreducible.
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