T/F. Give a counterexample if false. (a) There exists no surjective map from P t
ID: 3110120 • Letter: T
Question
T/F. Give a counterexample if false. (a) There exists no surjective map from P to R. (b) There exists no bijective map from P to R. (c) There exists a bijective map from P to Z. (d) For a set X and Y X, it is impossible for X and Y to have the same cardinality. (e) Let X and Y be finite with #X greaterthanorequalto #Y. Then there exists no surjective function from X to Y (f) Let X and Y be finite with #X greaterthanorequalto #Y. Then there exists no injective function from X to Y. (g) Let f: X rightarrow X be injective. Then f is surjective. (h) Let f: X rightarrow X be injective and let X be finite with n elements. Then f is surjective (i) Let n elementof Z. Then n|ab implies n|a or n|b. G) Suppose for a, b elementof Z that 3a + 5b = 1. Then g.c.d(a, b) = 1.Explanation / Answer
A) false as P is the power set hence cardinality of P is more than R
Example F:P-> R defined as F(x)=x then for every x in R there corresponds an element in P being power set if R.
B( true as it can never be ine one as cardinality of P is greater than R.
C) false let f(x) =x from P to Z then it is not one one as cardinality P is greater than cardinality of Z.
D) true if X and Y are uncountable .
E) false as if we define F : N-> {1,2} such that f(x) = 1 if x is odd and 2 if x is even . Then it is onto map.
F) True as cardinality of Y is less . Example in e) proves that.
G) false it is possible iff X is finite . Let f: N->N defned as f(×) =2x then for x=1 there does not exists any preimage.
H) true as if f is finite and injective from X to X then it is onto .
I) true as n|ab iff it divodes any one or both of them . Then only we can write ab as a multiple of n.
J) true by eucidean extended algorithm
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