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noof said:Hellohow are towdayyyyyy??
I have quistion(Whew)
and I hope you answer**************************
In each of the following two sets, A and B, are given. Answer the following questions about each
pair of them.
(a) Is A = B?
(b) Is A B?
(c) Is B A?
(d) Compare the cardinalities of A and B.
(i) A = P(P(P(;))) B = P(P(P(P(;))))
(ii) A = P(X [ Y ) B = P(X) [ PY )
(iii) A = P(X \ Y ) B = P(X) \ P(Y )
(iv) A = P(X Y ) B = P(X) P(Y )
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Let X be a set and let f1 and f2 be functions from A to R. For x 2 X let g(x) = f1(x) + f2(x) and
h(x) = f1(x)f2(x). Verify that g and h are functions.
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In class we showed that jZ+j = jNj and that jZj = jNj. Give a bijective function f : N ! Z to
show that jNj = jZj (i.e. cardinality of the set of natural numbers ([f0; 1; 2; 3; : : :g) is the same as the
cardinality of set of all integers).
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Using de nitions of the set operations show that if
X Y and X Z then X Y \ Z
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Express the following in terms of predicate logic (using nested quanti er and appropriately de ned
predicates).
(a) If jXj < jY j, then there can not be an onto function from X to Y .
(b) If jXj > jY j, then there can not be an one-to-one function from X to Y .
(c) Principle of mathematical induction is an important proof technique which works as follows:
Suppose we want to show that the predicate P is true for all positive integers n, we complete
two steps.
Basis step: Show that P is true for 1.
Inductive step: Show that for every positive integer k, if P is true for k then P is true
for k + 1.
(d) The principle of Well Ordering states that \every nonempty set of positive integers has a
minimum element".
(e) The Pigeon-hole Principle states that if n+1 pigeons are placed in n pigeon-holes then some
pigeon-hole must contain more than 1 pigeons. **************************************************************************************************
pleeeeeeez help me any Q ...
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