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Find whether the following statements ar...

Find whether the following statements are true or false. If the statement is false, then write its correct statement:
(i) If P={m,n} and Q={n,m},then `PxxQ={(m,n),(n,m)}`.
(ii) If A and B are non-empty sets, then `AxxB` is a non-empty set of the ordered pairs (x,y) such that `x in A and y in B`.

Text Solution

Verified by Experts

The correct Answer is:
N/a
(i) `:. P={m,n} and Q={n,m)`
`:. N(P)=2 and n(Q)=2`
`:. n(PxxQ)=2xx2=4,i.e., ` there are 4 ordered pairs in `(PxxQ)` while it is given that `(PxxQ)={(n,n)(n,m)}` i.e., there are 2 ordered pairs in `PxxQ`
Therefore given statement is false.
Now, `PxxQ={(m,n)(m,n)(n,n)(n,m)}`
(ii)If A and B are two non-empty sets then from the definition of cartesion product
`AxxB={(x,y):x in A. y in B)}`
Therefore, the given statement "If `AxxB` is a non-empty set of ordered pairs (,y) then `x in and y in` A'' false.
True statment is `: AxxB` is a non-empty set of ordered pairs, (x,y) then `x in A and y in B. `
(iii) To prove `: Axx(Bnnphi)=phi`
`:. Bnnphi` = set of common elements of set B and empty set `phi=phi`
`:. Axx(Bnnphi)=Axxphi=phi`
Therefore, given statement is true.
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