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Question

In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

(i) If xA and AB then xB

(ii) If AB and BC then AC

(iii) If AB and BC then AC

(iv) If A⊄B and B⊄C then A⊄C

(v) If xA and A⊄B then xB

(vi) If AB and xB then xA.

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Solution

(i) The statement is false.

Take A = {1}, B = {{1}, 2}

Now 1A and AB but 1B.

(ii) The statement is false

Take A = {1}, B = {1, 2}, C = {{1, 2}, 3}

Now AB and BC but AC.

(iii) The statement is true.

Let xAxB ( AB)

xC ( BC)

Now xA xC

AC.

(iv) The statement is false.

Take A = {1, 2} B = {2, 3 }, C = {1, 2, 5}

Now A⊄B and B⊄C but AC.

(v) The statement is false.

Take A = {1, 2} and B = {2, 3, 4, 5}

Now 1A and A⊄B but 1B

(vi) The statement is true.

Let xA xB ( AB)

Now xB xA.


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