A set T is defined recursively by 2 elementof T If x element

A set T is defined recursively by: 2 elementof T If x elementof T, then so are x + 3 and 2x. State whether each of the following numbers are in T: 6, 7, 19, 12, 14.

Solution

if 2 belongs to T then 4, 5 belongs to T {2,4,5}

if 4 belongs to T then 7,8 belongs to T {2,4,5,7,8}

if 5 belongs to T then 8,10 belongs to T {2,4,5,7,8,10}

if 7 belongs to T then 10,14 belongs to T  {2,4,5,7,8,10,14}

if 8 belongs to T then 11,16 belongs to T {2,4,5,7,8,10,11,14,16}

if 10 belongs to T then 13,20 belongs to T {2,4,5,7,8,10,13,14,16,20}

if 11 belongs to T then 14,22 belongs to T {2,4,5,7,8,10,14,16,22}

if 16 belongs to T then 19,32 belongs to T {2,4,5,7,8,10,14,16,19,22,32}

......

hence 7,19,14 belongs to T