Let A={x,a,b,c,d} a) How many closed binary operations f on A satisfy f(a,b)=c?

Question

Let A={x,a,b,c,d}
a) How many closed binary operations f on A satisfy f(a,b)=c?
b) How many of the functions f in part (a) have x as an identity element?
c) How many of the functions f in part (a) have an identity?
d) How many of the functions f in part (a) are commutative?

Explanation / Answer

a>There are 5*5=25 input pairs {5 choices for each operand of the binary operation}. Now each such pair except (a,b) has 5 choices of answer so 24 operand pair have 5 choices of answer...giving a total of 5^24 solutions. b>For x to be an identity element we need to fix the answers to 10 more operand pairs {5 of which have x as the first operand and 5 of which have x as the second operand} besides fixing the answer of the pair (a,b). So in total we have 5 choices for the remaining 25-11=14 pairs...so 5^14 solutions. c> If a function has an identity element then there can be only one such identity element ...now for x as an identity element there were 5^14 elements...also a and b cannot be identity elements because f(a,b)=c!=a!=b...so c and d can be the other identity elements ... so the total number is 3*(5^14) d>For the commutative property we need to only consider pairs of operands without their order being important which in total is 5C2=10. Out of there f(a,b)=f(b,a)=c is already fixed ...so we have 5 choices for the remaining 10-1=9 pairs..so 5^9 solutions..

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