It is easy to get that both R and S are contained in A×A, so R∪S is contained in A×A, and R ∴R∪S is also a binary relationship on A. 。
∫ar∪sb →( a, b) ∪ r ∪ s → (a, b) ∈ r or (a, b)∈S→aRb or BRA or aRb→(b, a)∈R or
∴R∪S is also symmetric.
2) Error
The fundamental mistake lies in the general situation of R.S.S.R.
Counterexample: Let A, B and C be different from each other, and r = {(a, A), (B, B), (C, C), (A, B), (B, a)}. s = {(a，a)，(b，b)，(c，.
(b, c), (c, b)}, it is easy to verify that both R and S are equivalent relations on {a, b, c}, and
R s = {(x, y) | t∈A, s.t.(x, t)∈R and (t, y) ∈ s} = {(a, a), (a, c), (.
S r = {(x', y') | t'∈A, s.t.(x', t')∈S and (t', y') ∈ r} = {(a, a), (a, b).
It is easy to get (a, c) ∈ R S, but (a, c) does not belong to S R；; ; So there is a, c∈A, which makes aR Sc but does not satisfy cR Sa,
So the synthesis of equivalence relation is not necessarily equivalence relation.
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【 Attachment 】 According to the definition of the synthetic relation R S, there are t∈A, s.t.(a, t)∈R and (t, b)∈S, and aR Sb iff exists.
The key problem lies in symmetry: let A, b∈A have Ar Sb → (a, b) ∈ r s → have T ∈ a, S T. (a, t) ∈ r and (T, b)∈S→ have T.
Transitivity should also be unsatisfactory under normal circumstances. If you are interested, find your own counterexample ~
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[[I won't write in detail later, give an idea, I can develop it myself slowly]]
3) Error
Counterexample: It is easy to verify that R={(a, b), (b, c), (c, a)} and S={(a, c), (c, b), (a, b)} are transitive relations on {a, b, c}, but (a, c)
4) Error
Look at the definition and you will know that it must be wrong. "A good order is a complete order with a good foundation." Obviously, the definition of a good order is stronger than that of a complete order, and it is a good order. A good order is not necessarily a complete order.
Counterexample: (z, ≤) is a complete order, but it is not a good order.
5) Correct
Of course, this proposition holds if you can give an example, which is similar to the meaning of 4).
Let a = "(x, ≤) be a good order "and b = "(x, ≤) be a complete order",
Then A→B, but there is no B→A, so a ≠ b.
6) Error
Let f: n→ n, f (0) = 0, f (n) = n-1(n >; 0),
g:N→N，g(m)= m+ 1；
It is easy to get that f g = I (n) is bijective, but G is not surjective, because 0∈N but 0 does not belong to g(N).
7) Correct
∵A is included in B,∴x∈A→x∈B,∴x∈A∪B→(x∈A or x∈B)→(x∈B or X ∈)
8) Correct
If a subset of a poset has a lower (upper) supremum, then the upper (lower) supremum is unique, which can be proved by reduction to absurdity. It is also very useful in the mathematical analysis of supremum mentioned here.
9) Error
The negation of the predicate "all yes" should be "not all yes", so the negation proposition should be "Zhang Ming and Li Hong are not both good students".
The difference between "nothing" and "not all" should be clear and semantically clear ~
10) correct
The definition of "reflexivity" comes out in one step (note that I(A)={(x, x)|x∈A}).
1 1) is correct.
It's too obvious ...
12) correct
Example: A = {1}, B = {1, {1}} are examples of satisfaction.
13) correct
I(A)∪I(B)=I(A∪B) can be directly proved from the conclusion of 10.
14) correct
(a 1，a2) r 2 (b 1，B2)，(b 1，B2) r 2 (c 1，C2) → (a 1，a2)，(b 1。
(b 1,b2),(c 1,c2)∈r→(a 1,a2),(c 1,c2)∈r→(a 1,a2)r^2(c 1,c2)
So as long as R is a relation, whether R is transitive or not, R 2 = R× R is transitive.
15) correct
Like 10), it is also a direct conclusion from the definition: iff (take A, b ∈ A has (A, b)∈R, (B, a)∈R→a=b) iff (take A, B ∈)
16) correct
It is easy to prove that {[a] | A | ARB} is the division of A with respect to R by taking the equivalence relation R on set A and a∈R, [a] =
That is, the equivalent class A/R about r.
17) error
See 2) for counterexamples.
18) correct
There are no conjunctions and punctuation marks.
19) correct
Know things clearly. The general method to prove the equality of sets is to prove bidirectional inclusion.
20) Error
This is an atomic proposition. "Red and yellow can be blended into orange" = "Red and yellow can be blended into orange",
There is also ≦ "Red can be turned into orange, and yellow can be turned into orange", which needs to be clarified.