I am supposed to use them to implement the function F(a,b,c) = ab' + a'c'.
I am aware of how to use a NAND gate. However, I have only used a demultiplexer to convert binary into decimal numbers.

Can some give me a hint on what I am supposed to be doing?

 

The way I did it is to expand the function into:

F = c'b'a'+cb'a+cb'a'+cb'a

F = 000+010+100+101

After that I connect the 3 input, A (LSB), B, C(MSB), into the 74LS138.

So when F = 1, one of the four output Y0,Y2,Y4 or Y5 will be 0.

From this you can get:

F = Y0'+Y2'+Y4'+Y5'

So using the output of the 74LS138, design a logic circuit using the NAND gates.

 

The way I do it is to expand the function into:

F = c'b'a'+c'b'a+c'ba'+cb'a

which is also the same as:

F = 000+001+010+101

So just put a, b and c into the select input of the 74LS138, where a is the LSB and c is the MSB. Then use the NAND gates to implement this function:

F = Y0'+Y1'+Y2'+Y5'

I did not check my work though, so there may be mistakes.

 

Well, you can implement this function (as any function) with the NAND only.
You can use the demultiplexer just as a selector between '1' and '0' inputs, and the nand gates output at the control...
You can of course implement it only with the demultiplexer - when a, b and c are at his control...(write a truth table)