Archangel Macsika
Write down the relation R on the Power-set of S = {1, 2, 3, 4}, defined by, R = {(A, B) | A and B are subsets of S and they have the same cardinality} using any one of the methods: (i) set of ordered pairs, (ii) directed graphs, (iii) matrix notation.
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Solution:
S = {1, 2, 3, 4}
R = {(A, B) | A and B are subsets of S and they have the same cardinality}
Subsets of S = {,{1},{2},{3},{4},{1,2},{1,3},{1,4},{2,3},{2,4},{3,4},{1,2,3},{2,3,4},{1,3,4},{1,2,4},{1,2,3,4}}
Now, we need those subsets whose cardinality are same/equal.
n()=0
n({1})=n({2})=n({3})=n({4})=1
n({1,2})=n({1,3})=n({1,4})=n({2,3})=n({3,4})=n({2,4})=2
n({1,2,3})=n({2,3,4})=n({1,3,4})=n({1,2,4})=3
n({1,2,3,4})=4
Thus, R = { ({1},{2}), ({1},{3}), ({1},{4}), ({2},{3}), ({2},{4}), ({3},{4}),
({1,2},{1,3}), ({1,2},{1,4}), ({1,2},{2,3}), ({1,2},{2,4}), ({1,2},{3,4}), ({1,3},{1,4}), ({1,3},{2,3}), ({1,3},{2,4}), ({1,3},{3,4}), ({1,4},{2,3}), ({1,4},{2,4}), ({1,4},{3,4}), ({2,3},{2,4}), ({2,3},{3,4}), ({2,4},{3,4}),
({1,2,3},{2,3,4}), ({1,2,3},{1,3,4}), ({1,2,3},{1,2,4}), ({2,3,4},{1,3,4}), ({2,3,4},{1,2,4}), ({1,3,4},{1,2,4}) }